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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Valuations of polynomial rings Macauley, Ronald Alvin 1951
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Title | Valuations of polynomial rings |
Creator |
Macauley, Ronald Alvin |
Publisher | University of British Columbia |
Date | 1951 |
Date Issued | 2012-03-09 |
Description | If R is a field on which all (non-archimedean) valuations are known, then all valuations on R[x], where x is transcendental over R , are also known. Ostrowski described such valuations of R[x] by means of pseudo-convergent sequences in the algebraic completion of A of R . MacLane later showed that if all valuations of R are discrete, then any valuation V of R [x] can be represented by certain "key" polynomials in R [x]. The present paper exhibits the connection between these two treatments. This is achieved by first determining keys for the valuation which a pseudo-convergent sequence defines on A[x], and then relating these keys to those for V . |
Subject |
Polynomials |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2012-03-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080629 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/41331 |
Aggregated Source Repository | DSpace |
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