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Valuations of polynomial rings Macauley, Ronald Alvin 1951

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Cef*\  VALUATIONS OF POLYNOMIAL RINSS by Ronald A l v i n Maeauley  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS  .We accept t h i s t h e s i s as conforming to the standard required from candidates f o r the degree of MASTER OF ARTS.  Members of. the Department «f *@thematics  -THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1951  Abstract If  R  i s a f i e l d on which a l l (nori-archimedean) valua-  t i o n s are known, then a l l valuations on transcendental over  Rfx^J, where  R , are also known.  x- i s  Ostrowski described  such valuations of R[xl by means of pseudo-convergent sequences i n the algebraic completion o f A  of  l a t e r showed that i f a l l valuations of R  are d i s c r e t e , then  any valuation  V  R .  MacLane  of R [x"] can be represented by c e r t a i n  "key" polynomials i n R [x}.  The present paper exhibits the  connection between these two treatments.  This i s achieved  by f i r s t determining keys f o r the valuation which a pseudoconvergent sequence defines xm these keys to those f o r V .  A T x ] , and then r e l a t i n g  A c k n o w l e d g m e n t  The writer wishes to express h i s thanks to Dr. B.N. Moyls of the Department of Mathematics at the University of B r i t i s h ColumMa f o r h i s advice and guidance.  His numerous c r i t i c i s m s and suggestions  proved invaluable i n the preparation of t h i s thesis.  1.  Introduction.  A non-archimedean valuation  V , hereafter  simply c a l l e d a valuation, of an integral domain single-valued mapping of the elements of numbers and  +  R  R  isa  Into the r e a l  such that:  0 0  1)  Va  2)  VO = + op ',  3)  V(ab) - Va + Vb  4)  V(a + b) > min fVa, Vb} f o r a l l a, b e R .  i s a unique f i n i t e r e a l number f o r a / 0 ,  f o r a l l a, b e R ,  An extremely Important property of these valuations i s that, if  Va ^ Vb , then  V(a +b) > Va,  V i a +b) = min {Va, Vb}; hence, i f  Va «« Vb .  Ostrowski, and l a t e r MacLane, attacked the problem of finding a l l extensions  of valuations on an integral domain  to the r i n g of polynomials over  R •  R f x],  where  x  R  i s transcendental  MacLane*s r e s u l t s are based on the assumption that  a l l valuations of  R  are discrete; t h a t i s , the r e a l numbers  used as values form an i s o l a t e d point set.  I t i s the purpose  of t h i s paper to provide a connection between the valuations of Ostrowski and MacLane on  R f x ] , where  R  i s a f i e l d with  only discrete valuations. Definition 1.1:  Let R be a f i e l d with a valuation  sequence { a ^ J , where with respect to i  V  V . The  e R , i s a pseudo-convergent sequence i f V(a^  - a^^) < V ( a  i + 1  - c^) f o r a l l  > N , so BE f i x e d p o s i t i v e integer. If {a.}  i s a pseudo-convergent sequence w i t h respect to  2 V , then the sequence /Vc^}  i s eventually s t r i c t l y monotone  increasing or eventually attains a constant value; as long as  0  i s not a limit of {<i±\, fVaJ-  converges to a f i n i t e limit.  This is important, for i t is essentially this property that Ostrowski uses to extend  V  on R to W  on RCxl, where W  is a valuation of R[x]. H shows ( t l ) , III, page 371) that e  if  f(x) e R[x], then  sequence.  {f[a±)}  This implies that  i s also a pseudo-convergent i s convergent to a  {Vfia^)}  f i n i t e limit except when {c^} converges to a root of f(x) . Hence, i f {a^}  i s a pseudo-convergent sequence possessing no  limit i n the algebraic completion  A  of R , the function  on, R[xl defined by .Wf(x) = lim Vf(a;i)  W  i s a valuation  i-»oo  ( [ i i , section 65, page 374) of R [ _ l . Further, ( [ l ] , IX, page 37$) every valuation of Rfx} maybe obtained by means of some pseudo-convergent sequence i n A • The pseudo-convergent sequences i n A are valued by an extension of V this extension always  exists([1"],  II, page  300)  on R  to A ;  • This last  reference implies that any valuation of R(x) may be extended to  A(x) . Hence, i f a l l valuations of A fx]  are found, a l l  valuations of R£x} are automatically found. This result i s of prime importance to the development of the theory i n this paper. Definition 1.2:  Let K be an integral domain with a valuation  V . Two elements written  a ~ b (V), i f V(a - b) > Va .  Definition 1.3: V  a, b e K are equivalent with respect to V ,  For a, b e K, a. equivalence divides  i f there exists  1  c e K such that  b in  b <~ ea (V) ; notation:  3.  a|b  (V) . If  V  i s any valuation of  discrete valuation  V  of  0  R[x]  R , MacLane ([2]) represents  by the following inductive method: i s assigned to f (x) - a x  a value  1 1  1  Q  shown to be a valuation of Vig(x) < vg(x)  f ( x ) £ R[x],  + ... + a , a f u n c t i o n  V-jf(x) » m i n f v ^ i + i ^ i j •  V  x  R fx} such that  f o r a l l g(x) e R[x].  V  , that i s ,  or there exists an  Vi  e  a  [  x  l  be  i s called V]x -  f o r a l l g(x) e  such that  92(x)  may  < V ; that i s ,  = £vo,  x  I f that l a t t e r i s the case,  R£x] i s  The value  V g(x) » Vg(x)  f (x) e R[x]  on  This function  a f i r s t stage value and i s symbolized by Either  V  Vjx = Vx = (i^  Then f o r any polynomial  + a^jx "  n  n  defined by  x •  which reduces to a  i s  • R[x},  V j f (x) < Vf (x) . chosen such that 92  i s a monic polynomial of. the smallest degree s a t i s f y i n g V^92  2 •  <  This polynomial s a t i s f i e s , over 1^ , MacLane's con-  d i t i o n s f o r a key polynomial. D e f i n i t i o n 1.4:  I*et  W  be any valuation of  Rfx] •  9 e Rfx] i s a key polynomial over .the value  nomial  (i)  9  W  A polyif:  i s equivalence i r r e d u c i b l e - 9|a(x)b(x)  implies either  9|a(x) (W)  or  9|b(x) (W)  (W) ,  (ii)  9  i s minimal - 9|a(x) (W) implies deg a(x) > deg 9 ,  (iii)  9  i s monic.  It i s shown ([2], theorem 4.2) that i f a key polynomial 9  over  W  W  on  R[xl  1  i s assigned  a value  jx = W 9 > W9 , then the  defined by  W»f(x) = min {wf (x) + i u j , i  function  4.  f(x) = f ( x ) 9  .where  + f _i(x) 9 ~  n  f i ( x ) < deg 9 , i s a valuation of and  Wf(x) < W*f(x)  +  n  + f (x),  R [x].  i f and only i f  Further,  on  ^2 *  v  2  >  V  l*2  *  T  h  e  v  a  l  u  e  and  V f ( x ) « V f ( x ) = Vf(x)  deg  f (x) < deg 92 •  by  2  0  R[x]  2  V 9 2 " ^2]*  A  V s>3 < V 9 3 . V2  nomial over  s  2  Again, i f  2  93  93  V  V  2  and  V2.  such that i s .symbolized  befbre, either V2 •» V of minimum degree s a t i s -  exists i t i s a key poly-  and may be used to define a v a l u a t i o n  that  V3 < V  with  deg f (x) < deg 93 •  and  V f ( x ) = V f ( x ) = Vf(x) 3  In  chosen above  2  The second-stage value  V^x • H I ,  .  i f assigned the value  satisfies  or there e x i s t s a monic polynomial fying  q>  f o r a l l f ( x ) e R[x}  x  v*2 = f V ,  <W  <p)f(x) (W); i n p a r t i c u l a r ,  the o r i g i n a l valuation V , the polynomial w i l l define a valuation V2  S  W  i f deg f ( x ) < deg 9 ( f 2 j , theorem 5.1)  Wf(x) = W»f(x)  V < p  d e  0  2  V3  such  f o r a l l f(x) e R[xl  The third-stage value  V3  i s sym-  bolized by  V3 • [ V , Yix = [ i i , V292 = |A2» V393 • u.3].  shows ( [ 2 ] ,  theorem 8,1) that i f t h i s procedure i s continued,  0  MacLane  equality w i l l occur a f t e r either a f i n i t e or countable number of steps*  In the f i r s t case V - V  k  - [V , V 0  1 X  V  w i l l have a representation  = M-1,V92 2  and i s called an inductive value*  88  ^ 2 » •••  1 k<Pk = ^k]» v  In the l a t t e r case  V = Tfco - [ v , V x » ( t V 92 = V2> ••• 1 k«Pk Pk> •••]» Voof(x) .* l i m V f ( x ) , and V i s c a l l e d a l i m i t value* v  o  where  x  l f  =  2  k  k-»°o  Hence each valuation of  Rfx]  two cases i f every valuation of  may be represented by one of these R  i s discrete*  The key polynomials d e f i n i n g the above inductive and l i m i t values s a t i s f y :  5" #iv)  ~ V±-l  i > (v)  i  s  f  a  ^  r  a  8  e  1  1  £  o  r  a  1  1  2 ;  deg ^  > deg 9 _ i  f  o  i -  i  Now since every valuation of R T x ]  2 •  i s e i t h e r an inductive  value or the l i m i t of a sequence of inductive values i t i s necessary, only to consider key polynomials which also s a t i s f y (iv) and (v) • For t h i s reason i t w i l l be assumed i n t h i s paper that a key polynomial  i s a polynomial  satisfying ( i ) , ( i i ) ,  (iii),  (iv) and (v) . The representation o f a valuation on  R [x]  i s not  necessarily unique, but i f one additional r e s t r i c t i o n i s placed on the key polynomials the representation becomes unique when V  0  on  R  i s discrete.  Let V = [ V , V-jx =• u^, ... , k  0  be an inductive valuation of R £x].  The  ?  V 9 ' • k  k  JAJ  value of  K  f ( x ) e R [x] i s found from the expansion f(x) - f ( x ) 9 *  f -l  +  n  where  deg f^(x) < deg 9  of  <Pk-l  of  ?k-2  a n c  .  k  (  x  )  ^k"  1+  U  -  +  o  f  By expanding each  (  x  >  )  f^(x) i n powers  * the c o e f f i c i e n t s of t h i s expansion again i n powers continuing these expansions, f i n a l l y the expression  f k ) . 2 where  n  aj e R  %  Y  t  J  2  .... &  ,  and each •ij <  g '1*1 deg 9  ,  d e  ±  i s obtained.  Furthermore,  Sow, the elements o f R  V f ( x ) « min {^ja^x k  l  j  9  2 2  j  . . .  9  k j k  j}.  may be p a r t i t i o n e d into classes of equi-  valent elements with respect to chosen from each class*  V  Q  and a representative may  In p a r t i c u l a r , the element  1' i s to  be chosen as a representative. These: representatives are c a l l e d the each  V -representatives.  I f i n the above expansion of  0  aj  mum value  is a  V -representative and a l l terms have the mini0  V f ( x ) , then k  Every polynomial  f(x)  i s c a l l e d homogeneous i n  f (x) e Rfx]  i s equivalent i n  only one homogeneous polynomial h(x)  f (x)  h(x).e R[x] i{2]  i s c a l l e d the homogeneous part of  f (x) •  \  V±~±  •  k  •  to one and lemma 16.2);  t  An inductive  or l i m i t value i s c a l l e d a homogeneous value i f each i s homogeneous i n  V  cp^, i £ 2 ,  MacLane has shown ([2], theorem 16,3  and 16.4) that any inductive or l i m i t value constructed from a discrete  V  0  may be represented by one and only one homogeneous  inductive or l i m i t value. The inductive and l i m i t values of  R[x] will'always be con-  sidered to be homogeneous values.  2. The r e l a t i o n between the valuations of Ostrowski and MacLane w i l l f i r s t be established on  A [ x j , where  A is an a l g e b r a i c a l l y  complete f i e l d . It w i l l be found convenient, i n t h i s section and future sect i o n s , to remove the condition that a MacLane value has f i r s t key x..  It is. necessary only that the f i r s t key be l i n e a r and monic.  The properties of MacLane values w i l l be preserved. Every valuation  V of  A[x]  may be defined by some pseudo-  convergent sequence {a:i}, with respect to  V  Q  on  A , which does  7*  not possess a l i m i t i n  A ;  Vf(x)  i s defined as  Vf (x) « l i m V f (.ai) . i-»oo • 0  These pseudo-convergent sequences may  be divided into two types.  To obtain the desired c l a s s i f i c a t i o n , a pseudo-limit i s defined. D e f i n i t i o n 2.1: An element  a e A  i s pseudo-limit of the  pseudo-convergent sequence {aj}, where to the valuation for  V  if  0  i > some integer  Note:  aje A , with respect  V ( a - a i ) - 8JL, where  1  $i <  0  N .  "Pseudo-limit" as defined here i s not the same as that defined by  Now  V (ai -  N'.  Since  0  a±+i)  Y'I  Ostrowski. = i * ^ , where )f± <  • o ^ i " i+l) v  a  a  follows that, f o r i > N ,  o[ji -  v  =  a  - _i  for  ^±+±  a) + (a  i > some integer  - a^i)]  it  .  The pseudo-convergent sequences are now  divided into two  classes: , (1) fai} possesses a pseudo-limit i n (2) {OL±\ Theorem 2.2: pect to  does not possess a pseudo-limit i n  0  V  of  A [x} V,  where  • lim  defined by  with r e s -  Ostrowski  i s the same- as the  first  ,  • l i m V (aj - aj ) . 0  + l  i-K»  ±-tOO  x - /3  a E A , then the  t  defined by' V, =fv 6 , V, (x - a) * ¥ \  stage v a l u a t i o n  Proof:  A ,  I f the pseudo-convergent sequence {o.±\  V , has a pseudo-limit  valuation  A ,  It i s s u f f i c i e n t to consider a monic l i n e a r polynomial in  A[x}.  V ( a i - a) - T G  = \ (a -  ±t  ), f o r  - /3  Since either i  V (a Q  ±  » (a^ - a) f: {a - / * ) and - ( 3 ) * J*i  sufficiently large.  or, V ^ . -  /3 )  a.  Hence V(x -f3)  = lim V ( a  i-*oo  Theorem 2.3: on  0  - /3 ) « min { T , V ( a - /* ) } « V ^ x  A  0  Given a finite inductive value  V « [Yo,V(x - a) = **]  A [ x j , a pseudo-convergent sequence \a^\  a e A  -/?)  with pseudo-limit  can be found such that r « l i m Yi  Proof:  Let  a / 0  in  A  be chosen such that  Then there exists a r e a l number sequence of integers {n±\  ni  o*  such that  no  i-*oo i  A  tit  2  l i m Iii-.  cd = Jf* .  can be found such that  -± < < ... < - i < 10 10 10 and  Va = d > 0 .  ...  1  m  a  1 G  Let  ft  be any one of the roots of  d = Va - V / ^ i  =10  Yft  or  x  - a  V ft = 1/10  .  d .  Then  Hence, the se-  quence { v / ^ " 4 } i s a s t r i c t l y increasing sequence with  lim Let  a  x  V ft*= lim £ 1 d = crd - Y  be defined by V(a; - a  •  ft  + a •  ) * V(/9i W i  i + l  Since  ft?)  i s a pseudo-convergent sequence and since a  i s a pseudo-limit of t h i s sequence.  quence {otjl  can have no l i m i t i n  defined by {<Xj}  .  - 1ft ,  {04}  1  V(<ij - a) = V ft ,  By Theorem 2.2, the se-  A , since the Ostrowski value  i s also defined by the f i n i t e value  V .  On combining Theorem 2.2 and Theorem 2.3 an equivalence i s obtained between valuations defined by pseudo-convergent sequences with pseudo-limits and the~inductive values of MacLane.  Theorem 2.4:  If  respect to  V,  value  V  » fv0,  X±  where  defined by {a^}  V  1  i s a pseudo-convergent sequence with  with no pseudo-limit  0  valuation  ^a^}  v  a  A , then the Ostrowski  i s the same as the MacLane l i m i t  V^x--!) » Xi  » ©^ i~ i+l^ •  in  , ... , V^x-a^) = X  sequence {a-gj i s pseudo-convergent with no l i m i t i n  Proof:  V  defined by  {o^|  i s equal to  A  V*  the  and  the  V* •  I f necessary remove a f i n i t e number of terms from the  beginning of the pseudo-convergent sequence {a$\ X±  the a's, so that v  ,  Also, i n a MacLane valuation  a  Ostrowski value  ...]  ±t  o^ i*n"^iJ a  **i  B  f  o  r  = V (ai-ai l) 0  i s s t r i c t l y increasing.  +  a 1 1  *  n  »  1  and renumber  V(x-a_.) = l i m ^ o ^ i + n ^ i ^  Since "  ^i»  rr*oo  Let  be defined-by  V  = [vo,  x  fines a f i r s t - s t a g e value of  V^x-ai) •  A [x]  ,  such that  then  Vj_ < V .  ^  de-  Now,  from  MacLane*s inductive argument used i n the introduction, i t follows immediately that  V  » [v0,  1  Vjtx-^) =  i s a MacLane valuation and also s a t i s f i e s  V  < V .  I f there  e x i s t s x - Z s A [x] such that V»(x - p ) < V(x - P = l i m V ( a j - p ) , then there e x i s t s a p o s i t i v e integer i-»oo 3  0  V'(x - P  that V»(x< V  a  ±  tion.  - P  )  for  valuation*  for a l l - P)}]  i > N .  V* - V .  From  x -  - a ,)' > V [x ±  i+1  Vi[(x - a and so  )  Hence,  « X± < l i m X i , f o r i-»oo  Thefefore  V-,^  - p  i  Q  - p )]  + {a  Q  = limfmin/ft, V (*±  ft)  i  ) < V (a  x - a  i  i + 1  +  )  i ~  ±  - (a  ±  i > N .  0  i + 1  ±  such  l i m Xi  - p  ) = V [(a 0  i + 1  -  a i  +  (a  V  i s a Mac_ane  T  -  i  i t follows that  f o r otherwise  HXjL+i) ] - V ( x - a ) ±  ±  -  Y  ±  > V (a 0  ±  -  - i + i ( i ) » which contradictsthe minimal a  v  )  i > N , "which is a contradic-  Suppose now that  » X,  ) N  Therljfore, from  it. follows that V (a  » (x - a^) - a)  , ...J  ... , V ^ x - a ^  a  i + 1  )  .10  - c o n d i t i o n ( i i ) of d e f i n i t i o n 1.4 Therefore &*i  Y (x - a ±  *i+l  <  i + 1  ) = Y  ±  and since  f° a l l i * 1 . 0  + 1  l {x - a ±  ) » V'[(x - a  i + 1  that  then  V$(a  : k  l i m V ia± i"*°°  - ft ) < » .  Q  - ft ) > Y± ,  ) < V  i + 1  (x-a  i + 1  ) ,  ) - (x - c^)] - Y i ;  hence f a i } i s a pseudo-convergent sequence. of {a±\,  i + 1  V^.  Now  r  V (ai - a i  f o r a key polynomial over  Let  I f ' ft were a l i m i t k > i  be chosen such  then  V ( a i - ft ) « V [ ( a i - a ) + ( a - ft )] = 0  0  k  k  Therefore V»(x - ft) = limlmin but  V»  i s a f i n i t e value.  V (a  - ft )}1 = l i m V (a.  ±  0  Hence {a±}  i  -ft)  = » ;  has no l i m i t i n A .and  w i l l , therefore, define an Ostrowski v a l u a t i o n which, by f i r s t part of theorem 2.4*. NOTE;  must be the same as  I f {a^} has a pseudo-limit  V* .  a e A , then  V»  may also  be represented by L  i-»oo  The r e s u l t s of t h i s section now provide a connection between the two methods of valuation  A fx].  In sections 4 and 5 i t w i l l be shown how a MacLane valuat i o n of  A[x] reduces to a MacLane valuation of  R Tx] , that i s ,  the key polynomials and t h e i r assigned values w i l l be found f o r the reduced valuation on a value on  R f x j to  valuation of be clear.  R fx}, and conversely how to extend  A fx] •  The connection between an Ostrowski  R Lx] and a MacLane valuation of  R Cxi w i l l then  -11.  3.  The key polynomials defining the r e s t r i c t i o n of a valuation  of  A f x ] to  R [x] are intimately related t o the key polynomials  used by MacLane ( [ 3 ] ) valuation  W  to extend a valuation  of R (a) ,  V  on  R  to a  .a; separable extension of  R  • For  0  t h i s reason a d e s c r i p t i o n of the methods used by MacLane and the essential r e s u l t s w i l l now be given. As a particular example,consider the inductive value v  of  R TxJ  k " [ o» v  l  v  Hl#2«P2  x  and reassign to  q>  ••• » Vk'Pk He]  e  18  the value  k  +  .  0 0  This defines a  new, generalized valuation V  =  of  [ V 1 V  X  2*2 • »*2»  =  V  »V l V l  R f x ] • The generalized valuation  V  88  hc-1*  V<p  k  tion  V  will  +«Q .. I f a  define a valuation  °°]  s a t i s f i e s a l l the con-  d i t i o n s of a v a l u a t i o n except that elements other than assigned the value  =  0  are  i s a root o f''<p , the valua(  W  on  k  R (a) . \ T h i s i s im-  mediately seen upon noticing that R (a)»fB5l and defining  W  by  Wf(a) = Vf(x) .  I f the  above are homogeneous i n the preceding has  y  it  for 2 < i < k ,  inductive value  shown.; ( [ 3 ] , theorem 5 . 3 ) that t h i s extension  the only extension of  V  0  to  W  V^-^, MacLane of  V  0  is  R (a) •  To f a c i l i t a t e - t h e discussion of the remainder of this  sec-  t i o n and i n view of sections 4 and 5 , i t i s convenient at t h i s point to define the terms projection and e f f e c t i v e degree. D e f i n i t i o n 3.1:  Let V  fc  "£  v 0  , ^ x » la^, ... , V q> = \xA be an k  k  12.  inductive value o f R [x] , I f n G(x) » g ( x ) 9k + g - l n  n  where deg gj^x) < deg 9 projection of  V  n—1 ?k~  ( x )  ,  k  proj (V ) • e - /3, where k  mum values r e s p e c t i v e l y of  ( x  i s a polynomial i n R [ x i , then the  with respect to  K  g© ) »  +  a  G(x) i s a -/2 , written  and ft are the maximum and minii  such that  V G(x) « V [ g i ( x ) 9k ] • k  k  D e f i n i t i o n 3.2: a  : written Let  of  V  0  is  k  D<pG(x) • a •  W  on  The e f f e c t i v e degree of G,(;x) i n Uj>o  be a valuation o f  R  and  a  R(a) , where  has minimal polynomial  W  i s an extension  G(x) e R[x].  By the  i isomorphism  R(a)c? R[x]/(G(x))  valuation  Vr en  valuation  V  ideal  Rfx_  i t i s c l e a r that a generalized  may be defined by  assigns the value  (G(x)) •  +  Vf|x) » Wf(a) . The  only to the members of the  0 0  I t would seem natural to construct  V  as MacLane  does f o r f i n i t e valuations; that i s , f o r valuations which assign the value V*l - [v *  +°Q l  v  o  Vi < V .  x  88  only t o  0 .  ^lj» wbe * 1  6  As before, a f i r s t - s t a g e value  V-l  e  V  x  f °° t  i  s  defined;  I t i s worth noting that proj (Vi) > 0 .  proj (V^) • 0  11  with minimum value, and  1 1  1  + ... + a  VG(x) = V^Gix) £  stage value a monic polynomial i s chosen.  f (x)  I f f (x)  0 0  9  2  •  •  0  To define a secon-  of minimal degree s a t i s f y i n g i s not homogeneous i n V j ,  then i t s homogeneous part i s to be chosen. part by  For i f  then would be only one term i n G(x) - anX .+ a ^ i x "  Vf(x) > Vjf(x)  again  Denote t h i s homogeneous  As was mentioned i n the introduction,  9  2  i s a key  13.  ^polynomial over by  V  and  • £ V , V-^ = u^, ^2^2  2  G(x)  • The second stage value  i s a homogeneous key over |i  b  0  2  ° .  That  i s then defined  2  p Vq> •  ^2]* where  88  q  V  2  93  , then  i  s  Now* i f  chosen as G(x)  G(x) i s a monic polynomial of minimal degree  satisfying  VG(x) > V ^ x ) w i l l follow from lemmas 3«3 and  Lemma 3.3:  Let V  !  be a k-th stage inductive value of  k  3.4.  RTx]  satisfying:  If  I  (1)  V f ( x ) < Vf(x)  (2)  deg f ( x ) < deg 9  (3)  \<?i - V<? = n ±  k  implies  V f ( ) - Vf (x) , k  x  for 1 < i < k .  i  i s a monic polynomial of minimal degree s a t i s f y i n g  V t<Vty k  Proof:  for a l l f ( x ) e R [ x l ,  k  , then  :  V f (x) < Vf (x)  implies  k  ^|f(x)  (V ) . k  Let f ( x ) have the quotient remainder expression f ( x )  = q^xty + r(x) , where deg r(x) < deg  • Then  V [ f - q t ] * - V [f - #] > minjvf, V[qfl}> min{v f, \ l q $ \ k  k  because of  (2), the choice of  q(x) . Hence Lemma 3.4: nomial  ty|f(x)  Let V  k  and the assumption (1) f o r  (V ) . ' k  be an inductive value of R[x] • Any poly-  G(x) £ R[x] has an equivalence decomposition G(x) ^ e(x)  9 ° k  t j  1  * 2  2  . . .  where each ty^ i s a homogeneous key over t£ > G  r  V  (V ) , k  k  , t > 0 and Q  f o r 1 < i < r , and e(x) i s an equivalence unit,  that i s , D^etx) = 0  ifl V  k  cept f o r equivalence u n i t s . Proof:  fl  Cf. [3J, theorem 4.2.  . This decomposition i s unique ex-  14. Now  q>2  suppose  *  s  homogeneous monic polynomial of mini-  a  ?2  ma}, degree s a t i s f y i n g  Vi<p  key over  V G(x) < VG , G(x) ~  ^  .  lemma 3.3. G(x) - cp • V G (x)  w h e n  '  Gfx)  homogeneous  a  l ^  v  2  G(x) ^ G(x) (V!) , and therefore by lemma  [v , V x -  o  is  h(x) <p ^  x  Hence i n t h i s case the value  2  If  Since  But  v  <  2  0  | i  x  l  V  3.4  i s given by  VG(x) » «>].  f  i s not a homogeneous key over  V^, then the second-  stage i s given by v  where  V  2  88  [ o» V  < V .  2  otherwise  v  l  x  ^1»  B  v  2?2  "  VG(x) ^ °° .  geneous key over some  V  [ o» v  v  l  x  G(x)  for  lemma  MacLane's  does become a homo-  or, i f t h i s does not occur, i t i s r e -  k  1*1 • •••  e  In the former case  • Vk  ° »*k»  by the preceding argument.  Also  9JL|G(X)  2 < i < k , and  proj  > 0  such that  ^  v  2  peated a countable number of steps. a  2  9 |G(X) ^ l ^  Also  inductive process i s repeated u n t i l  v  proj (V ) > 0  It i s noticed again that  V G ( x  (V^_I)  for  > - °°] for a l l i  1 < i < k .  If  a countable number of steps are required, then V = V« - [ v , V x « j i Q  Certainly, eaeh  9  k  \<  V  and  x  ... , V 9 k  l f  9 JG(x) ( v " k  k-1  = H, k  k  < deg G(X) ..  some point on a l l the keys w i l l have the same degree. case i t can be shown {[2], lemma 6.3) i s discrete i f the value group;; of r e a l numbers used as values f o r If  V f (x) < Vf (x)  for a l l  k > 0  f o r some  V  ...] •  for a l l k £ 2 .  )  , deg 9  i s minimal over  k  Since So from In t h i s  that the value group of &  Y  i s d i s c r e t e ; that i s , the  form an i s o l a t e d point set.  f (x) e E M ,  then  V f (x) < Vf (x) k  by the monotone increasing character of the i n -  f  1-5. ductive values and so for a l l k > 0 group of  V  1  <P |f(x) (V ) . k+1  (Cf. the introduction).  V f(x) < \ fc  +  f(  1  But sinee the value  i s discrete, Vf (x) > l i m V f (x) = k-*» k  Therefore only polynomials i n V<f(x) < Vf(x) V m V  Therefore  k  but since  oo .  (G(x))  could s a t i s f y  V(f (x). = » = Vf (x)  f o r ? ( x ) e (G(x)),  . It i s seen that every d i s c r e t e  to a f i n i t e separable extension  V  0  R(a)  of of  H E  may be extended by MacLane's induc-  t i v e process, where the homogeneous keys can be further r e s t r i c t e d to s a t i s f y the conditions proj (V^) > 0 <P.JJG(X)  of  a .  (V^^)  f o r i 2i 2 , where  for i > 0  G(x)  and  i s the minimal poly-  In faet, i t i s not d i f f i c u l t to see that these r e s t r i c -  tions are necessary. From the preceding arguments i t follows almost  immediately  that every such sequence of values constructed by these r e s t r i c t e d keys w i l l give a valuation of  R(a) •  The construction of such a sequence of values' may be accomplished i n a systematic manner. f i r s t - s t a g e value such t h a t  Let V^ - £ v o , V^x = a^J be a  proj (V ) > G  has been defined, the next key  1  for G(x) .  and each  The corresponding value u-i > Vj[_i«pi • Vi  V _ 1  1  <f>^ i s chosen as any one of the  tyj occurring i n the unique equivalence decomposition 3.4.  If  of lemma  i s chosen so that proj (V ) > 0 i  In the sequence of valuations so defined,  i s c a l l e d an i - t h approximant to  G(x) .  MacLane not  only shows that every such "sequence of values defines a valuation  *6.  W on  R(a)  which i s an extension of  (1) i f G(x) the  V  but that :  Q |  eventually becomes a homogeneous key over  inductive value  unique f o r  V , then the i - t h approximant i s k  2 < i < k  and also, the value  unique (f3] , theorem 5.3) . may be extended to theorem 10.1  R(a)  jx^  is  This implies that  i n only one way  V  0  ([3l ,  ) ,  (2) i f a countable sequence of keys are required, then there i s at most a f i n i t e number of d i f f e r e n t that can be constructed. tended to ([3l  R(a)  Hence,  V  0  on  R  sequences may be ex-  i n at most a f i n i t e number of ways  , theorem 10.1) . W of  4* The reduction, or r e s t r i c t i o n , of an inductive value A[xl  to  Rtx]  w i l l f i r s t be found; following theorem 4.7 the  reduction of a l i m i t value w i l l be found.  These r e s u l t s w i l l be  established by mathematical induction. Theorem 4.1: of  Afxl  Proof: W (a 0  - ft )  0  W(x - a) =)f] i s any inductive value  Wa > Y , then  with Let  W •• [*W,  If 0  x - ft e A[xl .  - W e  ft  ; and  W - W» - [w , 0  If  W (a - Z 0  3  Wx - Y]. f  ) < t,  .  then  *7. Let W • [ W  Theorem 4*3:  value of A f x ] V  where ¥  Q  • W  W(x - a) - X\  q >  Y > Wa  with  V  , then  Q  be an inductive = [v  X  V-jx = Wa]  o >  ,  0  on R , i s the f i r s t - s t a g e of the reduction of  Q  to R[x] •  There exist polynomials  f ( x ) e R f x ] such that  V f ( x ) < Wf(x) . x  Proof: fore  i s Wx • min { Y  The value of x  Wi = [w , Wjx * W a] Q  V-^ - £ V ,  reduction to  V-jX - W a]  Q  Rfx] . Let G(x) * fx - a) (x - Z^) i n Rfx} •  \ix  W (x - a) < W(x - a) and 1  ... (x -  ft ) t  Since  < W(x -  ±  ) ,  = W^Gfec) < mix) ..  Theorem 4.3 :shows that f o r Y > Wa  at least one more key  Q  i s necessary to obtain the correct reduction of Lemma 4.4:  Let W = fw , W(x - a) =Y]  A f x ] • A polynomial  by  x - a  in W  f(x) £ A f x l  be the expansion o f f (x)  |w f  the :re_ation  i s equivalence d i v i s i b l e  Q  n  ±  Q  + 1?},  n  + f _i(x - a) " 1 1  n  1  + ... + f  W f Q  G  0  > Wf(x) ; and because always holds.  Then  - W f(a) > Wf(x)  and, therefore, f ( x ) ~ f ( x - a )  in  x - a|f(x) i n W •  W ; that i s ,  W [ f (x) - {f_(x - a ) n  f (x) y  q(x)(x - a)  where  Wh(x) > Wf(x) •  i n W.  pansion o f h(x), i s f  Then  But, since  f  Q  Since  = f(a) ,  Suppose  W f(a) > Wf(x). 0  0  i n powers of x - a ; f ^ e A •  W f(a) > Wf(x)  0  to R f x l .  i f and only i f W f(a) > Wf (x) .  Let f ( x ) - f ( x - a )  Wf (x) « min  W  be any inductive value  p  of  Proof:  . There-  Q  i s the f i r s t - s t a g e of the  Q  be the minimal polynomial of a  V 3 G U )  0  i s a f i r s t - s t a g e value to ¥ ;  0  < W . Hence  W a}« Wa  t  n  + ... + f ( x - a)}] x  n  + ... + f ^ x - a )  Suppose, now,  fix)  = q(x) (x - a) +.h(x) ,  h , the l a s t term i n the ex0  i t follows that  18.  W f 0  = W h > Wh(x)  Q  o  > Wf(x)  0  ; that i s , W f(a) > Wf(x) . Q  In the r e s u l t s to follow the polynomials... <p^ numbers  and the r e a l  w i l l he the homogeneous key polynomials and t h e i r  values which are used by MacLane to extend a value to a value  on R(a) ( £ 3 ) . Since the value  W  Q  and therefore on Rfa) , i s given, of  W  V  extension of t h i s Theorem 4.5-s  V  to the given  0  Let the polynomials  W  0  q>i and \i± defining the W  on R(a) .  0  <pj and the numbers  to Rfr]  o  V  k  provided that ,  •  on R to  0  " [ o' V  V  l  x  x - aL.  "  V  in W  1  i s given by " h>>  2*2  V k  ± false for a l l i s  By Theorem 4 . 3 and lemma 4 . 4  Proof:  =  [ o» V  l  v  x  88  KL> ••• > V l ' P k - l  stage of the reduced value and f(x) e R f x ] .  i n the  V  k  V;  0  on R  to W  0  defined to be V _i£(x) k  Wf(x) ^ \_jf(x)  i  (k-Dst  s  for a l l  f ( x ) = W f(a) , because i n exQ  on R(a) the value  when  W f (a) i s 0  deg f ( x ) < deg <p • But, since k  by lemma 4 . 4 , i t i s concluded  Q  k  ^k-l]  then  be any polynomial such that  » W f(a) > Wf(x)  Wf(x) * V _ i f ( x )  deg f (x) < deg  k - 1  =  k - 1 ,  V^., f (x) < Wf(x)  Let f ( x ) e R,fx_  deg f ( x ) < deg <p . Then tending  ^  =  this theorem i s true  k « 1 • Suppose the r e s u l t i s t r u e up to  for  that  V  u^ be  1 <; i < k .  internal  k-l  f  on R(d') . The k&th stage of the reduction o f  W =[w, W(x - a)  v  on A  0  the keys and values which define the extension of W  on R  0  w i l l be the r e s t r i c t i o n  0  to R • However, there exist  0  V  f o r a l l f ( x ) e R Tx] such that  i. Iswever,  W9 = W-^Oa) • p, > V _ i 9 i k  k  k  t  19. since  x - a|cp  in W  k  next key over  V  i s false,  with value  k - 1  and <p  may then be chosen the  k  V <p = | i k  k  k  (Gf. introduction) .  Also, the value V  k  [ o» l  =  V  V  x  s a t i s f i e s the r e l a t i o n  88  H>  ••• » k - l V l 7  V f ( x ) < Wf(x) k  - **k-l» Vk  If  are the keys used to extend  W  on R(a) , then there e x i s t s an  in  W »  0  Proof:  [w,  i  ^k]  f o r a l l f (x) e R[xl and  i s , therefore, a k-th stage of the r e s t r i c t i o n o f W Lemma 4.6;  B  V  such that  to RTx]. on R to  0  x - &)<Pi  W(x - a) = y ] .  0  There are two cases to consider: (a) W  i s found by an inductive value - then the l a s t  0  key i s G(x) ., the minimal polynomial f o r Rfx], which i s d i v i s i b l e by x - a equivalence d i v i s i b l e by (b) W o i  x - a  a in  and, therefore  in W ,  i s found by a l i m i t value - i f there exists no such that  x - ajcp^  every  9j_, with value  of  to R[x] .  W  G(x)  i n W , then by theorem 4*5 ^  , occurs i n the reduction  But t h i s implies the value of WG(x) < + « J .  i s + op ; while  Lemma 4.6 implies the existence of a f i r s t key Vk+1 which i s equivalence d i v i s i b l e by x - a V  [V  k =  V  l  x  53  1*11  in W .  By theorem 4.5,  ••• » Vk " ^k]  i s the k-th stage of the reduction o f W  to R [ x ] . There are two  p o s s i b i l i t i e s f o r the k-th stage value of <p i , either k+  Vk+1 Theorem  4.7:  < %k+l  o  r  Let W =  Vk+1  fw  oi  58 w  ^k+l •  W(x-a) =  H  , with  ¥ > W a , be Q  .20.  given on  A[x] . Let {Vj.}  be the sequence of approximants to  G(x) , the minimal polynomial of V  0  on R t o ¥  on Ria) . I f 9  0  these approximants such that of  W to R[x]  o  when  V 9 k  k  w  value  k+  W9  88w  <¥  Sfgd i n (2)  k +  9 k + 1  ... , V 9 k  k  ^ ,9  k +  = u ] when k  i , may be chosen as the next key with V < W on R£xl . I f V, i n (1) and V  , the two r e s u l t s may be given  1  t x  deg f ^ x ) < deg 9  k + 1  of minimum degree such that  k + 1  f  o* ) x  ,  and f (x) e R [ x l , a monic polynomial V»f (x) < Wf (x) . Then  V f f ( x ) * V f (x) 0  V = V , t h i s i s immediate from the d e f i n i t i o n of  Vf (x) .  k  9k+l  Q  i s a key polynomial over  Vk+i«  V [ f ( x ) - g(x) 9  k + 1  ]  Suppose  V f ( x ) < V f ( x ) , then k  Q  k  - V f(x) > V f ( x ) k  V'gix) » W§(x)  k  D  and f ( x ) / v g ( x ) 9 Q  Q  ¥\f (x) = V ' f i x )  f o r then  k + 1  (V ) K  <P +i • Now, k  because deg g(x) < deg f i x ) ; therefore  V»[f(x) s f ( x ) ] = V U - g i x ) ) + V ' 9 e  If  Vk since i t defines  which contradicts the minimal condition of the key  That  +  n  0  an approximant  ]  deg f ( x ) < deg 9^^ . Since i n  « f ( x ) - g(x) 9  k  k  Suppose the existence of zi n 1 f i x ) = f ( x ) 9k+l f - l ) <Pk+l * •••  V» • Vk, then  k+1  0  +  If  » W9  k + 1  V f (x) - W f(a) = Wf<x) f o r a l l  n  where  k  ',  l f  are both denoted by  by one proof.  = |x , V 9  k  ¥k+l •  ; t h i s gives  k+1  i s the f i r s t key i n  x - a|?k+l(W) , then the reduction  VjX •» p  0  f (x) e Rfx] such that k  i  k  As i n theorem 4.5,  (1) V 9 -L < 9  +  » u i , ... , V 9  1 X  k + 1  V « [v, Vk+l  Proof:  k  i s given by:  (1) V = [v, V  (2)  a , defining the extension of  k + 1  » Wi-g(x)) + W9  may be seen by assuming  k+1  = w[fix)-f0l  V?f (x) >.V»f(x) ; tf  .21  - V ' f ( x ) = V»[f(x) -  f U)] 0  « w[f(x) -  f ( x ) ] > m i n { w f ( x ) , Wf (x)} 0  Q  >V«f(x) .  Now  w [ f ( x ) - g(x) 9k+l] = Wflx) > V'f(x) » V f f ( x ) » Wf (x) ; 0  hence  0  f ( x ) ~ g(x)<p (W) . 0  Since  k+1  x - a|f (x-)- (W) . 0  tradicts  But, since  x - a|q> (W), k+1  deg f ( x ) < deg <p Q  W f (a) • Wf (x) ; therefore Q  0  V  Q  1  - W  t h i s con-  k+1  on R[x] i n either  (1) or (2) . On combining theorems 4.2 and 4*7 and lemma 4*4 a picturesque description of the reduction can be given i n terms of the size of  J* .  on  Alx] with  Y  Y  < Wa c  , V£x = Y~\  o  t i o n increases t o than  W a , the key 92 0  minimal polynomial V  2  [ o» l  B  V£<p  V  2  |\  V  • |A  x  s  increases t o W<p  2  creases to V  2  increases to W_a • . When Y  V-jx = H i ]  °? the  **] •  the reduction i s  Q  A  the reduci s just larger  second appro xi mant to G^x) , the  of a , i s needed.  ^1* ^2^2  B  o l  Y < Wa  For  As Y  .  o  and examine the reduction to R t x l as  continuously increases.  V£ « [ v  W * £ w , W(x - a) = Y~\  For t h i s purpose consider  s  • W Q ^  ^ 0  The reduction i s increases again the value  ^  ~  ^2  a  n  d  ^  • [*V , V^x = J J - I , V292 = M-2^ •  reduction i n -  t h i s process i s  0  continued i t i s seen t h a t as. Y  e  increases the reduction sweeps  through the approximants t o G^x) which describe the of  V  0  to W  0  on R(a) .  duction and the corresponding  extension  The only difference between the r e approximant defined by the same keys  i s that the value assigned t o the l a s t key i n the reduction may be l e s s than i t s value i n the approximant.  But as  t h i s value w i l l increase t o the corresponding i f the l a s t key i s not  Y  i s increased  approximant value  G(x) • For then J* would have to increase t o »  22.  W is a  It w i l l now be shown what this reduction i s when l i m i t value.  In the remainder of this s e c t i o n  ¥  i s defined by  S  Y  W ( x - aj) =  W = [w , Q  x  ... , ¥ ( x - a )  l9  Y,  m  ±  i  ,  ±  where the pseudo-convergent seauence | a f | has no pseudo-limit i n A .  I t should be noticed that Wi » [ w  0  >  wi(x-  ca) - *Y,  . . . , w ( x - ai) = ^ i ] ±  W ^ ( x - a^) » ^ i ] .  can also be represented by ¥^ = [w , 0  W ^ f (x)  find  , f (x)  i s expanded i n powers o f x W^^.  c o e f f i c i e n t s are valued with  But the c o e f f i c i e n t s are i n Q  Lemma on  W - ¥ {  The value  ±  +  - [ w ,  1  W^+j-tx -  0  A[x] . Let x - ft e A _ x l .  Proof:  V _ + i - ft) a  w  +1  a  all  W (x  since  that i+1 i + 1  y  i i  ft  ft  N ¥  A  i  a i  (x  all i > 0 .  i - ft -ft  ^  -  ft  ,  )}  then  )} - ^  and  ) .  e Atxl there e x i s t s a p o s i t i v e i n -  exists, then ¥(x - ft ) > yj (x  - ft ) f o r  ±  < ¥  i + 1  .  Now  i + 1  )>  w 0  (a  > o ( i + l - Z ) • Hence, a  ( a  ¥ (a 0  i + 1  - ft ) > W ^ f x -  x - ft = (x - ai+i) + (ai+i - /* )  - ¥ (x - a  w  o  ¥(x - ft) = W^ (x - ft ) f o r a l l I > N .  I f no such  for otherwise from  = ¥  x -  N - such that  i> 0  0  -  ±  For each  ¥ (  w  then  minf^,  i + 1  0  - ft )  Proof:  (*i -ft)] -  +  I f ¥ (ai - ft ) >  ) > min f ^ ,  3  Lemma 4.9: teger  a  +  0  6^ i+l - Z ( x  a  ±  0  o[( i i - i)  = ¥i(x-/?) .  i+l  I f ¥ ( a i - ft ) < Y  = min/^, W ( a  ¥ £ ( x - P)  w  and the  and are therefore a c t u a l l y valued by W .  A  w  For, t o  3  i + 1  0  i+1  i t follows  and therefore  ¥(x - ft ) - ¥ ( a  ) , which contradicts Since ¥ ( a  - ft ) ;  0  i + 1  - /3 )  ¥(x - ft ) > ¥ ( x - ft ) f o r i  -/S)>¥ (x-/3)»¥ (x-a + ) = Y i  ft)  i  i  1  ±i  therefore  23.  ^  W (a Q  i + 1  - /3 ). W [ ( a 0  for a l l i > 0 .  i + 1  - a  1 + 2  ) + (a ft  But t h i s implies  i + 2  -/*)] = ^i+i  i s a psreudo-limit of  M. Theorem 4.10: of  k ( i ) keys occurring i n the reduction  Wj • [w#f'W|(x -  first  )  on  ATxl  [w , 0  to  W (x - a ) - ^ x  R  [ x l are the  R [x3  are the f i r s t  Wf (x) - Wjf (x)  the reduction of  Wj, then  V  w  , for 1 < v <k(i),  k(i) - 1  f(x) e R [x]  for a l l i > N .  W  o  value.  n  keys i n the reduction of  I t is. only  W^, +  of  i + 1  =[W , W G  i + 1  (x- a  i + 1  If V  ) »  k ( i )  is  i i i ,  necessary then are the f i r s t  and that the values, with  the possible exception of. Pk(i) » are the same. and W  there exists  R txl gives every poly-  to show that the k ( i ) keys i n the reduction of k(i)  values, f o r  f ( x ) = Wf(x) for a l l  kfi)  Hence, the sequence of values f V ^ ) } nomial i n R£x] the correct  u  W •  By lemma 4.9, -for any given  N . such that  • - •]  i  Also, the values  the keys i n the reduction of Proof:  , ... , W (x - c ^ )  x  i n the reduction of  an  A[x] to  k ( i ) keys i n the reduction of W =  on  The  By lemma 4.3, Wj_  define the same valuation  A [ x j • Hence, they w i l l have the same reduction on  R [x*]  and, because the keys i n the reductions are homogeneous, each reduction w i l l be i d e n t i c a l with respect to keys and values ([2], theorem 16.4) . to  As the value of x - <*i+i  the valuet l * ( i ) k  appearing i n V j j . ^ ^ ) v  k(i)  t  0  v  i s increased from  might increase and the keys, i f any,  but not i n V ^ d ) . are used to augment  k ( i + l ) * These are the only changes that can happen;  and at l e a s t one of these changes must happen.  The truth of  t h i s follows from the discussion immediately after theorem 4.7  .24  and the f a c t that the minimal polynomial of increases as the value of  x - ai+i  0^+1  definitely  increases.  Theorem 4.11: The reduction of a l i m i t value R[x], as described i n theorem 4*10,  W  Arx7 to  on  i s a l i m i t value.  Proof: Suppose the reduction i s an inductive value V  By lemma 4 . 9  * [ y , V^x = Hi, ... , V q> • |j, ] .  k  o  k  a smallest  i  such that  Wj_  to  cp  assumes the value  k  R[x3 must be  V  k  W^q> «= W<p = n k  •  k  •  k  there exists  The reduction of  since, t h i s i s the f i r s t stage i n which  k  n  and  k  minimal polynomial of < WG(x)  k  < W .  in  Since by assumption  But f o r  G(x) , the  RDcl, V G(x) » W^tx) k  V  k  < W  i+i  G(x)  = W , t h i s contradiction es-  tablishes the theorem. 5.  In section 4 the connection between a value  i t s reduction to  of  A£x] and  R£x] was established. The converse problem  w i l l now be solved; that i s , given a value an extension of  V  that a value  on  W  W  to  Atx3.  V  of  Rcxl , to f i n d  F i r s t , however, i t w i l l be shown  A Cxi may be w r i t t e n i n a standard form.  In the following theorem the. notation V  a  [ o» V  v  l  x  88  »*_'• ••• » V k  i s to mean that a l e a s t the keys up to resentation of keys past  V ; however,  <p , or k  7  V  "  ]  q>  occur i n the rep-  k  may be an inductive value with  may even be a l i m i t value.  This notation  w i l l also be used for  w = [w , w x 0  Lemma 5.1;  I f a value V_« | V ,  on  R£x], then  x  * ui,  W»  0  Vix = | i  V  may  of l t  ...^,-w(x , k  a\ k  =.jr, k  ATxl reduces to  ... , V  k 9 k  be extended to  = u- , k  ]  »  25. W =  [w , W-jx  on  A [x] where:  - [i  0  (1) (2) (3) (4)  ±  = W'(x  Wg/  W (x  - a)  2  = 0  <Pi (a )  Y±  l t  for  - ai)  = ¥'/3  3  Wf(x)  Vi - fv ,  /3 e A  W  on  0  and  t i o n of  k^  W]_  W,  f(x) e A[x] . W^(x  OJ  - a^) • for  be defined by  = |Tv,  i = 1, 2, ft  Wj_x = u^]  .  V-jx • u-jj •  o  1  ^i~\  ft - W»  0  0  ZTa^  f(x) =  W  Wj^ • £w,  i s certainly  , k ,  e A ,  ft  W± • [w  defined by  j  * *k»  a  1, 2, ...  i *  for a l l  A  " k^  x  ... , k ,  t  for a l l  w  i  Vix • ( i i , ... , Vi<pi =  0  Proof: Let  1, 2  for  > Wf(x)  •••  2  i =  Further, the reduction of is  Y,  88  2  ,  ... , k .  for The reduc-  Let  f(x) e A[xl,  i  then W'f(x) > min {w^  + iW'x}* W f(x) .  The value  s a t i s f i e s (2),  and also (1)  91 • x  (3),  x  (4)  Assume theorem true up to W 9  and  f  > V _^9  k  k  a factor  x - a  = W  k  ft  9  of  k  .  Then  W _i  ct^ = 0 .  and  reduces to  k  For a key to augment  W  V _i k  let  k-1  be chosen so t h a t W (x-a ) > W _i(x-a ) k  ft)  W(x - a ) £ W» (x k  such that  k  k-  x  1  k  (Gf. introduction) and x -  9  9  of  k  k - 1  W 2.*  since  W  k  k  f o r any f a c t o r  W' (x - ft ) > W ^ U  - ft ) .  Now  define  W by W » [w , W (x - a ) = Y = W (x - a ) ] . For x -ft , 1  k  k  k-1  9  any factor of  k  k  k  k  k  ,  min{Y ,  W (x  -  W (a  k  - ft ))  , and  W'(x  - ft) > m i n { y , W ( a  k  - ft )}  .  k  ft)  -  k  k  0  0  The inequality cannot hold; f o r , then, W(x -  ft ) > Y  k  - W (x k  which contradicts the choice of W» (x l 9 T  k  - /3) m w (x k  = W 9 k  k  .  -  Since  ft)  - a ) > W (x k  k  x - a . k  - ft ) > \ ^ ( x  ft ) ,  -  Therefore  for a l l f a c t o r s of  9k(<*k) • 0 , c e r t a i n l y  ^  ;  so  x - a |9 k  k  in  W  k  .  26. J h i s means the redaction w i l l use only keys But  Wk9k  Wcpk = pk > Wk-l9k ,hence the reduction of  =  f  must be  .  Theorem 5.2:  o  k  W» (x - ft) = s ]  W» = [ w ,  I f a value  reduces t o V • £ v ,  o  V^x • M-I» ••• » k^k ** ^k"] v  W* = W - [ w ,  W(x - a) «Y]  0  Proof:  Let V  where  a » a  k  W < W  f  .  R  Cxl ,  on A[xV as i n theorem 5*1}  I f there e x i s t s an x - © e A[x]  W(x - 9) < W»(x - Q) , then f o r  such that  n  Y=8 .  . Also  be extended to W  , then  k  o  on ACxl  <pfc(a) - 0 , and a Y so  then there e x i s t s an a , such that that  y± f o r i < k .  G(x) e R[x]  , where  G (©) = 0, i t follows that V G(x) = WG(x) < ¥»G(x) = V G(x)=; k  hence,  k  Y • W(x - < x ) = W ( x - a ) < 8  Also, since  -  ft )  - W(x  .  and  f  5 = W»(x  W = W  - ft) <Y,  Y " 8 .  then  Let W« = [ w ,  Theorem 5.3:  o  be a value on A £x] v  then  W*  which reduces t o  [ o» l  s  v  W^x -ft\)= S i , ... , W*(x-/^-Sj,.  v  x  ^1* ••• » V k  a  88  ^k> •••)  o  n  R t x l  >  may be represented by  W = £W , Wjx = q  W (x - a ) « Y 2  2  2  , ... , W (x - a ) • k  k  Y  k  ..  where: (1)  9 (a ) = 0  (2)  reduction of W  Proof:  k  k  f or a l l k > 1 , k  to R f x ] i s V  fc  ,  The proof i s s i m i l a r to that f o r theorem 5.2.  From theroems 5.2 and 5.3 i t i s seen that every valuation of A fx} may be put into a form such t h a t each  a  k  i s a root of the  J27. -corresponding key cp^ appearing i n the reduction of the valuaR [ x ] • This  t i o n to V  information indicates how a valuation  of Rtx] may be extended to some valuation of A fx] . I t  w i l l now be shown how t h i s extension can be accomplished. Lemma 5.4:  Let a  W • J w , W(x - a) • If] can be defined on  then a valuation A[x_  9 e Atx] ,  be a root of some polynomial o  9  such that  M> • The value J*  has a prescribed value  i s uniquely determined*  Let 9 = ft ( x - a)  Proof:  n  n  where ft± B k  t  W /  i  max JTi . i  For t h i s value of  Y  , with  T  I* - o ^ 1  +  W  W  Y  defined by  fore the value  t  m and  Jf,  and the equality holds  > J*' > with the desired property.  9(a) = G , there exists an i ^ 0  So  - a) ,  Suppose there were two values  f o r at least one value o f i • and  for i = 1 , 2,. . . .  • jx  + i J* > jx f o r i • 1 , 2 , ... , n  ?  Q  ft^ix  + ... +  0  Q  Y -  n—1  - a)  W9 = min | W ft ±+ i^}« Let the numbers Y±  then  be defined by W fti + lY^ defined by  ft  +  1  i y  >  W  oA  +1  Y  Since  such that  **' •  would give  9 a value  W9 < \i .  There-  W(x - a) i s unique.  Lemma 5.5: In an inductive value V  k = [ o» l V  V  13  "1> ••• » V l ^ k - l " *k-tfk*k  for 1 < i < k  the  proximant s t o 9  §3  - »k\  o  ) V  i f V 88  f o» V  l  R  W  •  k  i s defined by V  f  are the complete, and only, set of ap-  This follows immediately from [3] , theorem 5.3  Proof: of  x  x  s  —  » k-l*k-l = hc-l^k v  s  (Cf • end  .Theorem 5.6: then  V  k  Let W  * [v,  A  be an extension o f  V^x = H i , • •• » Vfc9k *  0  extended to Proof:  on  Q  W * £w,  W(x - a)  0  on  y/  =  Wcpk = Hk • f o» v  i  v  x  Now,  n  »  i s uniquely deterreduces to  *tr-i»  * ] > v  where J < k , by theorem 4.7  since  and, so, V^ 9  This implies yf • k  1  k  = M„  88  ^k^k •  That i s , the reduction of  9 (a). ? 0 •  W  a  where  and  v » Hk •  <pk 5 and,  V  on  k  Vk •  may be defined  9k (a) • 0 , i t i s seen that f o r a given  the maximum number of extensions of i s the degree of  lAtyk = Hk  i s the extension of  Since theorem 5.2 elaims every valuation W by some  But  k  i s \ar  W  9k(<*) = 0*  X  s  R ,  may be  A [ ] , where  W defined by t h i s  - ^i»  on  Q  Cxl  R  W(x - a) = Y  By lemma 5.4 the value  mined from  o  V  R Tx] to  W  W  0  on  A Cxi  every extension may be found by the  method of theorem 5.6 • Theorem 5.7: let of  Let W  on  Q  A  be an extension of  V •"£-V , Vpc * H I , ... , Vk9k  88  0  Rrx] . The value  V  H-k» •••]  V  R and  be a l i m i t value  may be extended to the MacLane value  W = ptf, Wjx = HI, ¥ (x - a ) = Y2> ••• » k ^ " k^ ^*k» ••• w  0  on  on  Q  2  x  a  =  2  ACx] where: (1)  ^(c^) = 0  (2)  f o r i = 1, 2, ... ,  i s uniquely determined by the f a c t o r and the value  (3)  P^(x - a ) ±  =» 1  Hi  of  x - ai  9i ,  in x - a  u  l  for a l l i > 1  (Cf. d e f i n i t i o n 3.2) , (4) Proof:  Wi  reduces to  Vi  on  R[x]  f o r i - 1, 2, ... .  It w i l l be possible t o formally construct the sequence of  values {Wi} i f i t can be. shown that there always exists a factor  29.  of  9^  satisfying  (3)  .  The construction of  with t h i s  factor maybe accomplished by the method of theorem 5.6. W defined by t h i s  ever, i t w i l l be necessary to show that 1  sequence of values is a c t u a l l y a MacLane value. that property W  value  (3) may  W  be defined over that  Dq,(x - a  (1), (2) and  (4)  in  W£  Y < y y "* of  k  , as  V  proj V  k-1  tinuously increase as x - a  °^  9k+l  may  be defined by  s  u  c  *hat  n  satisfies OD,  <p  > 0  k  i s expanded i n powers of  in  FC  k  if  <p i  •  = [w  tf]  (2), (3) and  must continuously i n <Pk+l*  So, i f  <p i  must also con-  k+  in  x - a  k  .  k +  9  - a  k + 1  )  =1  in  < W (a 0  that i s , J * < W ( a k  Q  w  then  W = W K  k+l  K + 1  k  x - a - a ^)  " k+l^ • - [ o» w  w  k  ^et  - ¥ (x  - a )  k  w  kfl(x - a  ; for, let  k + 1  k +  W ^  where  K+  It only remains to show that  k  = W (x k  ;  k+  a  k  ,Y  k  a  W i  Now  k+1  .  x -  W defined by t h i s sequence of values i s a MacLane value. D (x  9k+l  This could only happen i f  ) • 1  (4)  (4)  with  W i ( x - <* ) * ^k+l]  k >  can  (2) and  Hence, there e x i s t s a factor k +  K + 1  (3)  Then as  , the value of  k  D^(x - a ^ W  i  with respect to x - a  .  k +  such  i s expanded i n powers  k+  9  can  satisfies  .  k  i  (4) and  (1),  k  < W^q> * y, < n  k  Y "*Tk k  satisfy  K  Q  the value of  k  l ^ k + l 2: 1  e  k +  Then, since  W  L t  Therefore,  ¥-*- &*  crease, since  •  k  W = [W , WJ(x - a )  be defined by  » M- "* l^k •  <p  i > 1 .  but such that  k  x - a  W  (4) a value  i t follows by induction that property  • be s a t i s f i e d f o r a l l and l e t  (2), and  also s a t i s f y i n g ( l ) , (2) and  K  i ) =1  k +  In order to prove  be s a t i s f i e d i t w i l l be shown that, given a  which s a t i s f i e s (1),  K  How-  k+l k + 1  D e  )  x - ft e ACx]  defined by  = y] k  ,  , then  Since a  k + 1  )  30.  W ( k  w  - ft ) = min { Y  W [a  kf  x  k l > " ft > -  m  i  0  { Y , W (a  n  k  +  V < k 1 " ft ) ^ min { w ( a  0  0  Therefore  W^  Hence, Let  and  w k +  >  <p  k + 1  ) } but  -  k + 1  k  and  Q  k  - /3 )} > m i n { f , k  have the same reduction  i n order that  W i  W (a -/3)] 0  k  on Rrx] •  reduce to V ^ . Let  k +  k +  have the expansion  k  <p *  f  k  W w  n  ^  x  " k^ a  n+  ' * o{Vi m  n  k i*k *  f  +  n-l^  ~ k^ ""  x  a  *\}  1  j V i +  +  Let  ^  ft)]  - a ) , W (a  a  +  -  k  i  W  n  a  k i  n  (  + •••  1  f ^ ( x - a ) ,then  +  k  d  - k l •  x  a  +  }  i , which cannot be zero, be chosen such that a minimum term  i s actually obtained i n the second inequality; then Wf + i i * > W © 0*1 'k - V k  • W <D > W f +iW (Y . a ) k+l k ~ o ! k+l °V *  1  Therefore, since w  o  W w  ( a  k+1  k " k+l a  (x - a  o K  9  f  1^0, -  J  k + 1  B  i  n  y +l  ) =  - k+l> • k + l a  >  k  w  ( x  k+1  ( x  Q  k  - a  k +  i) • Y  k  k  a  w  ** £ k + l w  W  ( x  k  ( x  - k>} » a  ~ k>> a  a n d  >  s o  b  u  t  »  " k> ^ **k • a  and since  W (ak - otk+i) £ ^ 0  $ +i k  sequence {ai} i s pseudo-convergent. l i m i t since  v x  " k+l>> k + l  But i t i s known from above that W (a  1  > W <x;:- a ) . Now,  k  / k+l w  1  >  Y  k  . Therefore  for a l l  k  t1  the  The sequence has no pseudo-  reduces to a l i m i t value; otherwise  V  would be  an inductive value ({2} , theorem 16.4) • There can be no l i m i t for the sequence {aj.} i n A  since  V  i s a f i n i t e value; a l t e r -  natively, every l i m i t of pseudo-convergent sequence i s a pseudolimit.  Therefore, W  i s a MacLane valuation of A [x]  properties (1), (2), (3) and  satisfying  Because of theorem 5»3 W  of  Afx],  D<p(x - a-k+i) = 1  of a l i m i t value Theorem  5.7.  and since f o r every l i m i t value  V  of  R[xl to  in  x - a  k  , every extension  A Cxi may be found as i n  Bibliography 1. A. Ostrowski, Untersuchen zur arithmetischen Theorie der KBrper, Mathematische Z e i t s c h r i f t , v o l . 39 (1934),  pp. 269 -  404.  2 . S. MacLane. A construction f o r Absolute Values i n Polynomial Rings, Transactions of the American Mathematical Society, v o l . 40 ( 1 9 3 6 ) , pp. 363 - 3 9 5 . 3.  S. MacLane, A construction f o r Prime Ideals as Absolute Values of an Algebraic F i e l d , Duke Mathematical Journal, v o l . 2 ( 1 9 3 6 ) , pp. 4 9 2 - 5 1 0 .  

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