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Valuations of polynomial rings 1951

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VALUATIONS OF POLYNOMIAL RINSS by Ronald Alvin Maeauley Cef*\ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of MATHEMATICS .We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members of. the Department «f *@thematics -THE UNIVERSITY OF BRITISH COLUMBIA April, 1951 Abstract If R i s a f i e l d on which a l l (nori-archimedean) valua- tions are known, then a l l valuations on Rfx^J, where x- i s transcendental over R , are also known. Ostrowski described such valuations of R[xl by means of pseudo-convergent se- quences i n the algebraic completion o f A of R . MacLane later showed that i f a l l 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 f i r s t determining keys for the valuation which a pseudo- convergent sequence defines xm ATx], and then relating these keys to those for V . A c k n o w l e d g m e n t The writer wishes to express his thanks to Dr. B.N. Moyls of the Department of Mathematics at the University of British ColumMa for his advice and guidance. His numerous criticisms and suggestions proved invaluable i n the preparation of t h i s thesis. 1. Introduction. A non-archimedean valuation V , hereafter simply called a valuation, of an integral domain R i s a single-valued mapping of the elements of R Into the real numbers and + 0 0 such that: 1) Va i s a unique f i n i t e r e a l number for a / 0 , 2) VO = + op ', 3) V(ab) - Va + Vb for a l l a, b e R , 4) V(a + b) > min fVa, Vb} for a l l a, b e R . An extremely Important property of these valuations is that, i f Va ̂  Vb , then Via +b) = min {Va, Vb}; hence, i f V(a +b) > Va, Va «« Vb . Ostrowski, and later MacLane, attacked the problem of finding a l l extensions of valuations on an integral domain R to the ring of polynomials R f x], where x i s transcendental over R • MacLane*s results are based on the assumption that a l l valuations of R are discrete; that i s , the real numbers used as values form an isolated point set. It i s the purpose of this paper to provide a connection between the valuations of Ostrowski and MacLane on R fx] , where R is 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 V . The sequence {a^J, where e R , is a pseudo-convergent sequence with respect to V i f V(a^ - a^^) < V ( a i + 1 - c^) for a l l i > N , so B E fixed positive integer. If {a.} i s a pseudo-convergent sequence with respect to 2 V , then the sequence /Vĉ } is eventually strictly monotone increasing or eventually attains a constant value; as long as 0 is not a limit of {<i±\, fVaJ- converges to a finite 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]. He shows (t l ) , III, page 371) that i f f(x) e R[x], then {f[a±)} is also a pseudo-convergent sequence. This implies that {Vfia^)} is convergent to a finite limit except when {c^} converges to a root of f(x) . Hence, i f {a^} is a pseudo-convergent sequence possessing no limit in the algebraic completion A of R , the function W on, R[xl defined by .Wf(x) = lim Vf(a;i) is 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 in A • The pseudo-convergent se- quences in A are valued by an extension of V on R to A ; this extension always exists([1"], II, page 300) • 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 is of prime importance to the development of the theory in this paper. Definition 1.2: Let K be an integral domain with a valuation V . Two elements a, b e K are equivalent with respect to V , written a ~ b (V), i f V(a - b) > Va . Definition 1.3: For a, b e K, a.1 equivalence divides b in V i f there exists c e K such that b <~ ea (V) ; notation: 3. a|b (V) . If V is any valuation of R[x] which reduces to a discrete valuation V 0 of R , MacLane ([2]) represents V by the following inductive method: a value Vjx = Vx = (i^ is assigned to x • Then f o r any polynomial f(x) £ R[x], f (x) - a nx n + a ^ j x 1 1 " 1 + ... + a Q, a function Vi on R£x] i s defined by V-jf(x) » m i n f v ^ i + i ^ i j • This function may be shown to be a valuation of R fx} such that < V ; that i s , Vig(x) < vg(x) for a l l g(x) e R[x]. The value i s called a f i r s t stage value and i s symbolized by = £ v o , V]x - • Either V x V , that i s , V xg(x) » Vg(x) for a l l g(x) e R[x}, or there exists an f (x) e R[x] such that Vjf (x) < Vf (x) . If that latter i s the case, 92(x) e a [ x l i s chosen such that 92 i s a monic polynomial of. the smallest degree satisfying V^92 < 2 • This polynomial satisfies, over 1^ , MacLane's con- ditions for a key polynomial. Definition 1.4: I*et W be any valuation of Rfx] • A poly- nomial 9 e Rfx] i s a key polynomial over .the value W i f : (i) 9 i s equivalence irreducible - 9|a(x)b(x) (W) implies either 9|a(x) (W) or 9|b(x) (W) , ( i i ) 9 i s minimal - 9|a(x) (W) implies deg a(x) > deg 9 , ( i i i ) 9 i s monic. It i s shown ([2], theorem 4.2) that i f a key polynomial 9 over W i s assigned a value jx = W 9 > W9 , then the function W1 on R[xl defined by W»f(x) = min {wfi(x) + i u j , 4. .where f(x) = f n ( x ) 9 + f n _ i ( x ) 9 ~ + + f 0 ( x ) , d e S fi(x) < deg 9 , i s a valuation of R [x]. Further, W <W and Wf(x) < W*f(x) i f and only i f <p)f(x) (W); i n particular, Wf(x) = W»f(x) i f deg f(x) < deg 9 ( f2 j , theorem 5.1) . In the original valuation V , the polynomial q>2 chosen above w i l l define a valuation V2 on R[x] i f assigned the value ^2 * V < p 2 > V l * 2 * T h e v a l u e v 2 satisfies V 2 V and and V 2f(x) « V xf(x) = Vf(x) for a l l f(x) e R[x} such that deg f (x) < deg 92 • The second-stage value V2. i s .symbolized by v*2 = f V 0 , V^x • H I , V 292 " ^2]* A s befbre, either V2 •» V or there exists a monic polynomial 93 of minimum degree satis- fying V2s>3 < V 9 3 . Again, i f 93 exists it i s a key poly- nomial over V2 and may be used to define a valuation V3 such that V3 < V and V 3f(x) = V 2f(x) = Vf(x) for a l l f(x) e R[xl with deg f (x) < deg 93 • The third-stage value V3 i s sym- bolized by V3 • [V 0, Yix = [ i i , V292 = |A2» V393 • u.3]. MacLane shows ( [ 2 ] , theorem 8,1) that i f t h i s procedure i s continued, equality w i l l occur after either a f i n i t e or countable number of steps* In the f i r s t case V w i l l have a representation V - V k - [V 0, V 1 X = M-1,V292 8 8 ^ 2 » ••• 1 vk<Pk = k̂]» and is called an inductive value* In the latt e r case V = Tfco - [ v o , V xx » ( t l f V292 = V2> ••• 1 vk«Pk = Pk> •••]» where Voof(x) .* lim V kf(x), and V is called a limit value* k-»°o Hence each valuation of Rfx] may be represented by one of these two cases i f every valuation of R is discrete* The key polynomials defining the above inductive and limit values satisfy: 5" #iv) ~ V±-l i s f a ^ 8 e £ o r a 1 1 i > 2 ; (v) deg ̂  > deg 9 i _ i f o r a 1 1 i - 2 • Now since every valuation of R T x ] is either an inductive value or the limit of a sequence of inductive values i t i s necessary, only to consider key polynomials which also satisfy (iv) and (v) • For this reason i t w i l l be assumed i n this paper that a key polynomial is a polynomial satisfying ( i ) , ( i i ) , ( i i i ) , (iv) and (v) . The representation of a valuation on R [x] is not necessarily unique, but i f one additional restriction is placed on the key polynomials the representation becomes unique when V0 on R is discrete. Let Vk= [V 0, V-jx =• u^, ... , V k 9 k ' • J AJ be an inductive valuation of R £x]. The ? K value of f(x) e R [x] is found from the expansion f(x) - f n ( x ) 9 * + f n - l ( x ) ^k" 1 + U - + f o ( x ) > where deg f^(x) < deg 9 k . By expanding each f^(x) i n powers of <Pk-l a n c* the coefficients of this expansion again i n powers of ?k-2 continuing these expansions, f i n a l l y the expression f k ) . 2 Y % t 2 J .... & , where aj e R and each • i j < d e g '1*1 , deg 9 ± i s obtained. Furthermore, V kf(x) « min {^ja^x l j 9 2 2 j . . . 9 k k j j } . Sow, the elements of R may be partitioned into classes of equi- valent elements with respect to V Q and a representative may chosen from each class* In particular, the element 1' i s to be chosen as a representative. These: representatives are cal led the V 0-representatives. If i n the above expansion of f (x) each aj is a V 0-representative and a l l terms have the mini- mum value V k f ( x ) , then f(x) i s called homogeneous i n V k • Every polynomial f (x) e Rfx] is equivalent i n \ to one and only one homogeneous polynomial h(x).e R[x] i{2]t lemma 16.2); h(x) i s cal led the homogeneous part of f (x) • An inductive or l imit value is called a homogeneous value i f each cp^, i £ 2 , i s homogeneous i n V±~± • MacLane has shown ([2], theorem 16,3 and 16.4) that any inductive or l imit value constructed from a discrete V 0 may be represented by one and only one homogeneous inductive or l imi t value. The inductive and l imi t values of R[x] will'always be con- sidered to be homogeneous values. 2. The relat ion 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 algebraically complete f i e l d . It w i l l be found convenient, i n this section and future sec- t ions, 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 inear 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 limit i n A ; Vf(x) is defined as Vf (x) « lim V 0 f (.ai) . i-»oo • These pseudo-convergent sequences may be divided into two types. To obtain the desired classification, a pseudo-limit i s defined. Definition 2.1: An element a e A i s pseudo-limit of the pseudo-convergent sequence {aj}, where aje A , with respect to the valuation V 0 i f V 0(a-ai) - 8JL, where $i < 1 for i > some integer N . Note: "Pseudo-limit" as defined here is not the same as that defined by Ostrowski. Now V 0 ( a i - a±+i) = i * ^ , where )f± < ^±+± for i > some integer N'. Since Y'I • v o ^ a i " a i + l ) = vo[jai - a) + (a - a ^ i ) ] i t follows that, for i > N , - _ i . The pseudo-convergent sequences are now divided into two classes: , (1) fai} possesses a pseudo-limit i n A , (2) {OL±\ does not possess a pseudo-limit i n A , Theorem 2.2: If the pseudo-convergent sequence {o.±\ t with res- pect to V 0, has a pseudo-limit a E A , then the Ostrowski valuation V of A [x} defined by is the same- as the f i r s t stage valuation V, defined by' V, =fv6 , V, (x - a) * ¥\ , where • lim • lim V0 (aj - aj + l ) . ±-tOO i - K » Proof: It is sufficient to consider a monic linear polynomial x - /3 i n A[x}. Since - /3 » (a^ - a) f: {a -/* ) and V G(ai - a) - T±t either V Q (a ± - ( 3 ) * J*i or, V ^ . - /3 ) = \ (a - ), for i sufficiently large. a. l i m I i i - . m a Hence V(x -f3) = lim V 0( a A - /3 ) « min { T, V 0(a - /* ) } « V^x -/?) i-*oo Theorem 2.3: Given a finite inductive value V « [Yo,V(x - a) = **] on A [xj , a pseudo-convergent sequence \a^\ with pseudo-limit a e A can be found such that r « lim Yi Proof: Let a / 0 i n A be chosen such that Va = d > 0 . Then there exists a real number o* such that cd = Jf* . A sequence of integers {n±\ can be found such that ni no tit -± < < ... < - i < ... 10 10 2 10 1 and i-*oo 1 G i Let ft be any one of the roots of x - a . Then d = Va - V/^i =10 Yft or V ft = 1/10 d . Hence, the se- quence { v / ^ " 4 } i s a s t r i c t l y increasing sequence with lim V ft*= lim £ 1 d = crd - Y . Let ax be defined by • ft + a • Since V(a; - a i + l ) * V(/9i W i- ft?) - 1ft1, {04} i s a pseudo-convergent sequence and since V(<ij - a) = V ft , a i s a pseudo-limit of this sequence. By Theorem 2.2, the se- quence {otjl can have no limit i n A , since the Ostrowski value defined by {<Xj} 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 ^a^} i s a pseudo-convergent sequence with respect to V 0, with no pseudo-limit i n A , then the Ostrowski valuation V defined by {a^} is the same as the MacLane limit value V1 » f v 0 , V^x--!) » Xi , ... , V^x-a^) = X±t ...] , where X± » v©^ ai~ ai+l^ • Also, in a MacLane valuation V* the sequence {a-gj i s pseudo-convergent with no limit i n A and the Ostrowski value V defined by {o^| is equal to V* • Proof: I f necessary remove a f i n i t e number of terms from the beginning of the pseudo-convergent sequence {a$\ and renumber the a's, so that X± = V 0 ( a i - a i + l ) i s s t r i c t l y increasing. Since v o ^ a i * n " ^ i J B * * i f o r a 1 1 n * 1 » V(x-a_.) = lim ^ o ^ i + n ^ i ^ " ^i» rr*oo Let be defined-by V x = [ v o , V^x-ai) • , then ^ de- fines a first-stage value of A [x] such that Vj_ < V . Now, from MacLane*s inductive argument used i n the introduction, i t follows immediately that V1 » [ v 0 , Vjtx-^) = ... , V ^ x - a ^ , . . . J i s a MacLane valuation and also satisfies V < V . If there exists x - Z 3 s A [x] such that V»(x - p ) < V(x - P ) = lim V 0(aj- p ) , then there exists a positive integer N such i-»oo that V'(x - P ) < V Q ( a i - p ) for a l l i > N . Therljfore, from V»(x- ft) = limfmin/ft, VQ(*± - P)}] it. follows that lim Xi < V a i - P ) for i > N . Hence, V 0 ( a i + 1 - p ) = V 0 [ ( a i + 1 - a i ) + {a± - p )] « X± < lim X i , for i > N , "which is a contradic- i-»oo tion. Thefefore V* - V . Suppose now that VT i s a Mac_ane valuation* From x - » (x - a^) + (a i - i t follows that V-,^ - a i + 1,)' > V±[x - a±) » X±, for otherwise Vi[(x - a i + 1 ) - (a ± HXjL+i) ] - V ±(x - a ±) - Y± > V 0 ( a ± - a i + 1 ) and so x - a i + i ~ - a i + i ( v i ) » which contradictsthe minimal .10 -condition ( i i ) of definition 1.4 for a key polynomial over V^. Therefore Y±(x - a i + 1 ) = Y± and since l±{x - a i + 1 ) < V i + 1 ( x - a i + 1 ) , &*i < *i+l f° r a l l i * 1 . Now V 0 ( a i - a i + 1 ) » V'[(x - a i + 1 ) - (x - c^)] - Y i ; hence fai} i s a pseudo-convergent sequence. If ' ft were a limit of {a±\, then lim VQia± - ft ) < » . Let k > i be chosen such i"*°° that V$(a k : - ft ) > Y± , then V 0 ( a i - ft ) « V 0 [ ( a i - a k) + (a k - ft )] = Therefore V»(x - ft) = limlmin V (a± - ft )}1 = lim V 0(a. i -ft) = » ; but V» i s a f i n i t e value. Hence {a±} has no limit in A .and w i l l , therefore, define an Ostrowski valuation which, by f i r s t part of theorem 2.4*. must be the same as V* . NOTE; If {a^} has a pseudo-limit a e A , then V» may also be represented by L i-»oo The results of this section now provide a connection be- tween the two methods of valuation A f x ] . In sections 4 and 5 i t w i l l be shown how a MacLane valua- tion of A[x] reduces to a MacLane valuation of R Tx] , that i s , the key polynomials and their assigned values w i l l be found for the reduced valuation on R fx}, and conversely how to extend a value on R fxj to A fx] • The connection between an Ostrowski valuation of R Lx] and a MacLane valuation of R Cxi w i l l then be clear. -11. 3 . The key polynomials defining the restriction of a valuation of A fx] to R [x] are intimately related to the key polynomials used by MacLane ( [3 ] ) to extend a valuation V 0 on R to a valuation W of R (a) , .a; separable extension of R • For this reason a description of the methods used by MacLane and the essential results w i l l now be given. As a particular example,consider the inductive value vk " [ vo» v l x Hl#2«P2 e ••• » Vk'Pk18 He] of R TxJ and reassign to q>k the value + 0 0 . This defines a new, generalized valuation V = [ V V1 X = V2*2 • »*2» » V l V l 8 8 hc-1* V < pk = °°] of R fx] • The generalized valuation V satisfies a l l the con- ditions of a valuation except that elements other than 0 are assigned the value +«Q .. If a i s a root of (''<pk , the valua- tion V w i l l define a valuation W on R (a) .\This i s im- mediately seen upon noticing that R (a)»fB5l and defining W by Wf(a) = Vf(x) . If the yit for 2 < i < k , above are homogeneous in the preceding inductive value V^-^, MacLane has shown.; ( [ 3 ] , theorem 5 .3) that this extension W of V 0 i s the only extension of V 0 to R (a) • To facilitate-the discussion of the remainder of this sec- tion and i n view of sections 4 and 5 , i t is convenient at this point to define the terms projection and effective degree. Definition 3.1: Let Vfc " £ v 0 , ̂ x » la^, ... , Vkq>k = \xA be an 12. inductive value of R [x] , If n n—1 G(x) » g n(x) 9k + g n - l ( x ) ?k~ + g© ( x) » where deg gj^x) < deg 9 k , i s a polynomial i n R [ x i , then the projection of VK with respect to G(x) i s a -/2 , written proj (V k) • e - /3, where a and ft are the maximum and mini- mum values respectively of i such that VkG(x) « V k[gi(x) 9k ] • Definition 3.2: The effective degree of G,(;x) in Uj>ko i s a : written D<pG(x) • a • Let W be a valuation of R(a) , where W i s an extension of V 0 on R and a has minimal polynomial G(x) e R[x]. By the i isomorphism R(a)c? R[x]/(G(x)) i t i s clear that a generalized valuation Vr en Rfx_ may be defined by Vf|x) » Wf(a) . The valuation V assigns the value + 0 0 only to the members of the ideal (G(x)) • It would seem natural to construct V as MacLane does for f i n i t e valuations; that i s , for valuations which assign the value +°Q only to 0 . As before, a first-stage value V*l - [vo* v l x 8 8 ^lj» wbe1*6 V-l e V x f °° t i s defined; again Vi < V . It i s worth noting that proj (Vi) > 0 . For i f proj (V^) • 0 then would be only one term i n G(x) - anX11 .+ a ^ i x 1 1 " 1 + ... + a 0 with minimum value, and VG(x) = V^Gix) £ 0 0 • To define a secon- stage value a monic polynomial f (x) of minimal degree satisfying Vf(x) > Vjf(x) is chosen. If f (x) i s not homogeneous in Vj, then i t s homogeneous part i s to be chosen. Denote this homogeneous part by 9 2 • As was mentioned i n the introduction, 9 2 i s a key 13. ^polynomial over • The second stage value V 2 is then defined by V 2 • £ V q , V-^ = u^, ^2^2 8 8 ̂ 2]* where p Vq>2 • Now* i f G(x) i s a homogeneous key over , then 93 i s chosen as G(x) and | i 2 b 0 ° . That 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! 3.4. Lemma 3.3: Let V k be a k-th stage inductive value of R T x ] satisfying: (1) V k f ( x ) < Vf ( x ) for a l l f ( x ) e R [ x l , (2) deg f(x) < deg 9 k implies V k f ( x ) - Vf (x) , (3) \<?i - V<?± = n i for 1 < i < k . If I i s a monic polynomial of minimal degree satisfying V k t < V t y , then: V k f (x) < Vf (x) implies ^|f(x) (V k) . Proof: Let f(x) have the quotient remainder expression f(x) = q^xty + r(x) , where deg r(x) < deg • Then V k [ f - qt]*- V [f - #] > minjvf, V[qfl}> min{v kf, \ l q $ \ because of (2), the choice of and the assumption (1) for q(x) . Hence ty|f(x) (V k) . ' Lemma 3.4: Let V k be an inductive value of R[x] • Any poly- nomial G(x) £ R[x] has an equivalence decomposition G(x) ̂  e(x) 9 k ° t j 1 * 2 2 . . . flr (V k) , where each ty^ is a homogeneous key over V k , t Q > 0 and t£ > G for 1 < i < r , and e(x) is an equivalence unit, that i s , D^etx) = 0 ifl V k . This decomposition i s unique ex- cept for equivalence units. Proof: Cf. [3J, theorem 4.2. 14. Now suppose q>2 * s a homogeneous monic polynomial of mini- ma}, degree satisfying Vi<p2 < v ?2 w h e n ' Gfx) is a homogeneous key over ^ . Since VxG(x) < VG , G(x) ~ h(x) <p2 ̂ v l ^ lemma 3.3. But G(x) ̂  G(x) (V!) , and therefore by lemma 3.4 G(x) - cp2 • Hence i n this case the value V i s given by V o [ v 0 , V xx - | i l f VG(x) » «>]. If G (x) i s not a homogeneous key over V^, then the second- stage i s given by v 2 8 8 [Vo» v l x B 1̂» v 2?2 " where V 2 < V . It is noticed again that proj (V 2) > 0 for otherwise VG(x) ^ °° . Also 9 2 |G(X) ^ v l ^ ^ lemma MacLane's inductive process is repeated until G(x) does become a homo- geneous key over some V k or, i f this does not occur, i t i s re- peated a countable number of steps. In the former case v a [vo» v l x e 1*1 • ••• • V k ° »*k» V G ( x > - °°] by the preceding argument. Also 9JL|G(X) ( V ^ _ I ) for a l l i such that 2 < i < k , and proj > 0 for 1 < i < k . If a countable number of steps are required, then V = V« - [ v Q , V xx « j i l f ... , V k 9 k = Hk, ...] • Certainly, \< V and 9 kJG(x) (v" k - 1) for a l l k £ 2 . Since eaeh 9 k i s minimal over , deg 9 k < deg G(X) .. So from some point on a l l the keys w i l l have the same degree. In this case i t can be shown {[2], lemma 6.3) that the value group of Y f i s discrete i f the value group;; of & is discrete; that i s , the real numbers used as values for V form an isolated point set. If V f (x) < Vf (x) for some f (x) e E M , then V k f (x) < Vf (x) for a l l k > 0 by the monotone increasing character of the i n - 1-5. ductive values and so <P k + 1|f(x) (V k) . Therefore V f cf(x) < \ + 1 f ( for a l l k > 0 (Cf. the introduction). But sinee the value group of V1 i s discrete, Vf (x) > lim V k f (x) = oo . k-*» Therefore only polynomials i n (G(x)) could satisfy V<f(x) < Vf(x) but since V(f (x). = » = Vf (x) for ?(x)e (G(x)), V m V . It i s seen that every discrete V 0 of H may be extended to a f i n i t e separable extension R(a) of E by MacLane's induc- tive process, where the homogeneous keys can be further restricted to satisfy the conditions proj (V^) > 0 for i > 0 and <P.JJG(X) (V^^) for i 2i 2 , where G(x) is the minimal poly- of a . 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 restricted keys w i l l give a valuation of R(a) • The construction of such a sequence of values' may be accom- plished i n a systematic manner. Let V^ - £ v o , V^x = a^J be a first-stage value such that proj (V 1) > G for G(x) . If V 1_ 1 has been defined, the next key <f>̂  i s chosen as any one of the tyj occurring i n the unique equivalence decomposition of lemma 3.4. The corresponding value i s chosen so that proj (V i) > 0 and u-i > Vj[_i«pi • In the sequence of valuations so defined, each Vi i s called an i-th 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 V Q | but that : (1) i f G(x) eventually becomes a homogeneous key over unique for 2 < i < k and also, the value jx̂  i s unique (f3] , theorem 5.3) . This implies that V 0 may be extended to R(a) i n only one way ([3l , theorem 10.1 ) , (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 different sequences that can be constructed. Hence, V 0 on R may be ex- tended to R(a) i n at most a f i n i t e number of ways ([3l , theorem 10.1) . 4* The reduction, or re s t r i c t i o n , of an inductive value W of A[xl to Rtx] w i l l f i r s t be found; following theorem 4.7 the reduction of a limit value w i l l be found. These results w i l l be established by mathematical induction. Theorem 4.1: If W •• [*W0, W(x - a) =)f] i s any inductive value of Afxl with W0a > Y , then W - W» - [w0, Wfx - Y]. . Proof: Let x - ft e A[xl . I f W0(a - Z 3 ) < t, then W0(a - ft ) - We ft ; and the inductive value V k, then the i-th approximant i s *7. Theorem 4*3: Let W • [W q > W(x - a) - X\ be an inductive value of A f x ] with Y > WQa , then V X = [vo> V-jx = W0a] , where V Q • WQ on R , is the first-stage of the reduction of ¥ to R[x] • There exist polynomials f(x) e R fx] such that V x f ( x ) < Wf(x) . Proof: The value of x i s Wx • min { Yt W 0 a } « WQa . There- fore Wi = [wQ, Wjx * W0a] i s a first-stage value to ¥ ; < W . Hence V - ^ - £ V Q , V-jX - WQa] i s the first-stage of the reduction to Rfx] . Let G(x) * fx - a) (x - Z )̂ ... (x - ftt) be the minimal polynomial of a i n Rfx} • Since W1(x - a) < W(x - a) and \ix < W(x - ±) , V 3 G U ) = Ŵ Gfec) < mix) .. Theorem 4.3 :shows that for Y > WQa at least one more key is necessary to obtain the correct reduction of W to Rfxl . Lemma 4.4: Let W = fwp, W(x - a) =Y] be any inductive value of A fx] • A polynomial f(x) £ A f x l i s equivalence divisible by x - a i n W i f and only i f WQf(a) > Wf (x) . Proof: Let f(x) - f n ( x - a ) n + f n _ i ( x - a ) 1 1 " 1 + ... + f 0 be the expansion of f (x) in powers of x - a ; f^ e A • Since Wf (x) « min |wQf± + 1?}, W Qf 0 > Wf(x) ; and because f Q = f(a) , the :re_ation WGf(a) > Wf(x) always holds. Suppose W0f(a) > Wf(x). Then W [ f (x) - {f_(x - a ) n + ... + f x ( x - a)}] - W0f(a) > Wf(x) and, therefore, f(x) ~ f n ( x - a ) n + ... + f^x-a) in W ; that i s , x - a|f(x) i n W • Suppose, now, f (x) y q(x)(x - a) in W. Then fix) = q(x) (x - a) +.h(x) , where Wh(x) > Wf(x) • But, since h 0, the last term in the ex- pansion of h(x), i s f i t follows that 18. W 0f Q = W oh 0> Wh(x) > Wf(x) ; that i s , WQf(a) > Wf(x) . In the results to follow the polynomials... <p̂  and the r e a l numbers w i l l he the homogeneous key polynomials and their values which are used by MacLane to extend a value V 0 on R to a value WQ on R(a) ( £ 3 ) . Since the value W0 on A f and therefore on Rfa) , i s given, V 0 w i l l be the r e s t r i c t i o n of W0 to R • However, there exist q>i and \i± defining the extension of this V 0 to the given W0 on R(a) . Theorem 4.5-s Let the polynomials <pj and the numbers u^ be the keys and values which define the extension of V0 on R to W0 on R(d') . The k&th stage of the reduction of W =[wo, W(x - a) to Rfr] is given by Vk " [ Vo' V l x " V 2 * 2 " h>> V k = ^ provided that x - aL. i n W ± s false for a l l i i n the , • 1 internal 1 <; i < k . Proof: By Theorem 4.3 and lemma 4 . 4 this theorem is true for k « 1 • Suppose the result i s true up to k - 1 , then v k - l = [ Vo» v l x 8 8 KL> ••• > Vl'Pk-l = ^k-l] i s (k-Dst stage of the reduced value and V̂ ., f (x) < Wf(x) for a l l f(x) e R f x ] . Let f(x) e R,fx_ be any polynomial such that deg f(x) < deg <pk . Then V k - 1 f ( x ) = WQf(a) , because i n ex- tending V;0 on R to W0 on R(a) the value W0f (a) i s defined to be Vk_i£(x) when deg f(x) < deg <pk • But, since Wf(x) ^ \_jf(x) » WQf(a) > Wf(x) by lemma 4 . 4 , i t i s concluded that Wf(x) * V k_if(x) for a l l f(x) e R Tx] such that deg f (x) < deg i. Iswever, W9 k = W-^Oa) • p,k > V k _ i 9 i - t 19. since x - a|cpk in W i s false, and <pk may then be chosen the next key over V k - 1 with value Vk<pk = | i k (Gf. introduction) . Also, the value Vk = [ Vo» V l x 8 8 H> ••• » 7 k - l V l - **k-l» Vk B ^k] satisfies the relation V kf(x) < Wf(x) for a l l f (x) e R[xl and i s , therefore, a k-th stage of the restriction of W to RTx]. Lemma 4.6; If are the keys used to extend V 0 on R to W0 on R(a) , then there exists an i such that x - &)<Pi in W » [w0, W(x - a) =y] . Proof: There are two cases to consider: (a) W0 is found by an inductive value - then the last key is G(x) ., the minimal polynomial for a i n Rfx], which i s divisible by x - a and, therefore equivalence divisible by x - a in W , (b) W is found by a limit value - i f there exists no o i such that x - ajcp^ in W , then by theorem 4*5 every 9j_, with value ^ , occurs in the reduction of W to R[x] . But this implies the value of G(x) i s + op ; while WG(x) < + « J . Lemma 4.6 implies the existence of a f i r s t key Vk+1 which is equivalence divisible by x - a i n W . By theorem 4.5, Vk = [V V l x 5 3 1*11 ••• » Vk " ̂ k] i s the k-th stage of the reduction of W to R [x]. There are two possibilities for the k-th stage value of <pk+i , either Vk+1 < %k + l o r Vk+1 5 8 w^k+l • Theorem 4.7: Let W = fwoi W(x-a) = H , with ¥ > WQa , be .20. given on A[x] . Let {Vj.} be the sequence of approximants to G(x) , the minimal polynomial of a , defining the extension of V 0 on R to ¥ 0 on Ria) . If 9 k + i is the f i r s t key in these approximants such that x - a|?k+l(W) , then the reduction of W to R[x] i s given by: (1) V = [vo, V 1 X » ui , ... , V k9 k = |xk, V9 k + 1 » W9k+1] when V k 9 k + 1 < ¥ 9 k + 1' , (2) V k « [v0, VjX •» p l f ... , V k9 k = u k] when V k + l 88 w¥k+l • Proof: As in theorem 4.5, V kf (x) - W0f(a) = Wf<x) for a l l f (x) e Rfx] such that deg f(x) < deg 9^^ . Since i n (1) Vk9k+-L < w 9 k + ^ , 9 k + i , may be chosen as the next key with value W9k+1 ; this gives V < W on R£xl . If V, i n (1) and Sfgd i n (2) are both denoted by V 1 , the two results may be given by one proof. Suppose the existence of zi n 1 fix) = f n ( x ) 9k+l + f n - l t x ) <Pk+l * ••• + fo* x) « f 0 ( x ) - g(x) 9 k + 1, where deg f ^ x ) < deg 9 k + 1 and f (x) e R[xl, a monic polynomial of minimum degree such that V»f (x) < Wf (x) . Then Vff 0(x) * V f (x) If V = V k, this is immediate from the definition of Vf (x) . If V» • Vk, then 9k+l i s a key polynomial over Vk since i t defines an approximant Vk+i« Suppose V k f Q ( x ) < V kf(x) , then V k [ f Q ( x ) - g(x) 9 k + 1] - V kf(x) > V k f D ( x ) and f Q(x)/vg(x) 9 k + 1(V K) which contradicts the minimal condition of the key <Pk+i • Now, V'gix) » W§(x) because deg g(x) < deg fix) ; therefore V»[f(x) s f Q ( x ) ] = VU-gix)) + V'9 k + 1 » Wi-g(x)) + W9k+1 = w [ f i x ) - f 0 l That ¥\fe (x) = V'fix) may be seen by assuming V?ftf (x) >.V»f(x) ; for then .21 -V'f(x) = V»[f(x) - f0U)] « w[f(x) - f 0(x)] > min{wf(x), WfQ(x)} >V«f(x) . Now w[f 0(x) - g(x) 9k+l] = Wflx) > V'f(x) » V f f 0 ( x ) » Wf0(x) ; hence f 0(x) ~ g(x)<pk+1(W) . Since x - a|q>k+1(W), x - a|f0(x-)- (W) . But, since deg f Q ( x ) < deg <pk+1 this con- tradicts WQf (a) • WfQ(x) ; therefore V 1 - W on R[x] in 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* . For t h i s purpose consider W * £ w o , W(x - a) = Y~\ on Alx] with Y < Wca and examine the reduction to R t x l as Y continuously increases. For Y < WQa the reduction i s V £ « [ v o , V£x = Y~\ . As Y increases to W_a • the reduc- tion increases to | \ o l V-jx = Hi] . When Y i s just larger than W0a , the key 92 °? the second appro xi mant to G^x) , the minimal polynomial of a , is needed. The reduction i s V2 B [Vo» V l x B ^1* ̂ 2̂ 2 s **] • A s ^ increases again the value V£<p 2 • |A increases to W<p2 • W Q ^ 0 ^ ~ 2̂ a n d ^ e reduction in- creases to V 2 • [*V0, V^x = J J - I , V292 = M-2^ • this process i s continued i t i s seen that as. Y increases the reduction sweeps through the approximants to G^x) which describe the extension of V 0 to W0 on R(a) . The only difference between the re- duction and the corresponding approximant defined by the same keys is that the value assigned to the last key in the reduction may be less than i t s value in the approximant. But as Y i s increased this value w i l l increase to the corresponding approximant value i f the last key i s not G(x) • For then J* would have to increase to» 22. It w i l l now be shown what this reduction is when W i s a limit value. In the remainder of this section ¥ i s defined by S W = [ w Q , W x ( x - aj) = Yl9 ... , ¥ i(x - a ± ) m Y±, , where the pseudo-convergent seauence |af| has no pseudo-limit i n A . It should be noticed that Wi » [ w 0 > w i ( x - ca) - *Y, . . . , w ± ( x - ai) = ̂ i ] can also be represented by ¥^ = [w0, W ^ ( x - a^) » ^ i ] . For, to find W ^ f (x) , f (x) i s expanded i n powers of x - and the coefficients are valued with W ^ ^ . But the coefficients are in A and are therefore actually valued by W Q . Lemma The value W ± - ¥ { + 1 - [ w 0 , Ŵ +j-tx - on A[x] . Proof: Let x - ft e A _xl . I f ¥ 0 (ai - ft ) < Y± then V a _ + i - ft) w o [ ( a i + i - a i ) + (*i -ft)] - w o ( a i - ft ¥£ + 1(x - P) = min/^ , W 0 ( a i + 1 m i n f ^ , - f t )} = ¥i(x-/?) . If ¥ 0(ai - ft ) > ^ , then w 6 ^ a i + l - Z 3 ) > min f ^ , ¥ 0 ( a i - ft )} - ^ and w i + l ( x - ft ) W±(x - ft ) . Lemma 4.9: For each x - ft e Atxl there exists a positive i n - teger N - such that ¥(x - ft) = W^ (x - ft ) for a l l I > N . Proof: If no such N exists, then ¥(x - ft ) > yj±(x - ft ) for a l l i > 0 since ¥ A < ¥ i + 1 . Now ¥ 0 ( a i + 1 - ft ) > W ^ f x - ft) for otherwise from x - ft = (x - ai+i) + (ai+i - /* ) i t follows that y i i - ¥ i(x - a i + 1 ) > w 0 ( a i + 1 - ft ) ; and therefore i+1 > w o ( a i + l - Z3 ) • Hence, ¥(x - ft ) - ¥ 0 ( a i + 1 - /3 ) = ¥ i + 1 ( x ) , which contradicts ¥(x - ft ) > ¥ i(x - ft ) for a l l i > 0 . Since ¥ 0(a i + 1-/S)>¥ i(x-/3)»¥ i(x-a i+ 1) = Y±i therefore 23. ^ W Q(a i + 1 - /3 ). W 0[(a i + 1 - a 1 + 2 ) + ( a i + 2 - / * ) ] = ^ i + i for a l l i > 0 . But this implies ft i s a psreudo-limit of M. Theorem 4.10: The k(i) keys occurring i n the reduction of Wj • [w#f'W|(x - ) on A[x] to R [ x l are the f i r s t k(i) keys i n the reduction of W = [w0, Wx(x - a x) - ̂  , ... , Wi(x - c ^ ) • - • ] on ATxl to R [x3 Also, the values u w , for 1 < v < k ( i ) , i n the reduction of are the f i r s t k(i) - 1 values, for the keys i n the reduction of W • Proof: By lemma 4.9, -for any given f(x) e R [x] there exists an N . such that Wf (x) - Wjf (x) f o r a l l i > N . If V k ( i ) i s the reduction of Wj, then V k f i ) f(x) = Wf(x) for a l l i i i , Hence, the sequence of values fV^)} o n R txl gives every poly- nomial i n R£x] the correct W value. It is. only necessary then to show that the k(i) keys in the reduction of are the f i r s t k(i) keys i n the reduction of Ŵ +, and that the values, with the possible exception of. Pk(i) » are the same. By lemma 4.3, Wj_ and W i + 1=[W G, W i + 1(x- a i + 1 ) » define the same valuation of 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 identical with respect to keys and values ([2], theorem 16.4) . As the value of x - <*i+i i s increased from to the valuet l* k(i) might increase and the keys, i f any, appearing i n Vjj.^^) but not i n V^d). are used to augment v k ( i ) t 0 vk(i+l) * These are the only changes that can happen; and at least one of these changes must happen. The truth of this follows from the discussion immediately after theorem 4.7 .24 and the fact that the minimal polynomial of 0^+1 definitely increases as the value of x - ai+i increases. Theorem 4.11: The reduction of a limit value W on Arx7 to R[x], as described i n theorem 4*10, is a limit value. Proof: Suppose the reduction is an inductive value V k * [ y o , V^x = Hi, ... , Vkq>k • |j,k] . By lemma 4 . 9 there exists a smallest i such that Ŵ q>k «= W<pk = n k • The reduction of Wj_ to R[x3 must be V k since, this is the f i r s t stage i n which cpk assumes the value nk and < W . But for G(x) , the minimal polynomial of i n RDcl, VkG(x) » W^tx) < W i + iG(x) < WG(x) • Since by assumption V k = W , this contradiction es- tablishes the theorem. 5. In section 4 the connection between a value W of A£x] and i t s reduction to R£x] was established. The converse problem w i l l now be solved; that i s , given a value V of Rcxl , to find an extension of V to Atx3. First, however, i t w i l l be shown that a value W on A Cxi may be written in a standard form. In the following theorem the. notation V a[ Vo» v l x 8 8 »*_'• ••• » V k " ] i s to mean that a least the keys up to q>k occur in the rep- resentation of V ; however, V may be an inductive value with keys past <pk , or 7 may even be a limit value. This notation w i l l also be used for w = [w0, wx  * u i , ...̂ ,-wk(x , ak\ =.jrk, » Lemma 5.1; If a value W» of ATxl reduces to V_« | V 0 , Vix = | i l t ... , V k 9 k = u- k, ] on R£x], then V may be extended to 25. W = [w0, W-jx - [ i l t W2(x - a2) 8 8 Y2, ••• i w k ^ x " ak^ * *k» j on A [x] where: (1) <Pi (a±) = 0 for i = 1, 2t ... , k , (2) Y± = W'(x - ai) for i * 1, 2, ... , k , (3) Wg/3 = ¥'/3 for a l l ft e A , (4) Wf(x) > Wf(x) for a l l f(x) e A[x] . Further, the reduction of W± • [wOJ W^(x - a^) • ^i~\ i s Vi - f v 0 , Vix • (i i , ... , Vi<pi = for i = 1, 2, ... , k . Proof: Let W0 on A be defined by W0 ft - W» ft for /3 e A and W]_ defined by Wĵ  • £w0, Wj_x = u^] . The reduc- tion of W, i s certainly = |Tvo, V-jx • u-jj • Let f(x) = ZTa^ 1 , f(x) e A[xl, i then W'f(x) > min {w^ + iW'x}* Wxf(x) . The value Wx satisfies (2), (3), (4) and also (1) since 9 1 • x and ct̂  = 0 . Assume theorem true up to W k-2.* Then Wk_i reduces to V k _ i and W f 9 k > V k _ ^ 9 k = W k - 1 9 k . For a key to augment Wk-1 l e t a factor x - a k of 9 k be chosen so that W1 (x-a k) > W k_i(x-a k) (Gf. introduction) and W(x - a k) £ W» (x - ft) for any factor x - ft of 9 k such that W' (x - ft ) > W^U - ft ) . Now define Wk by Wk » [wk-1, Wk(x - a k) = Y k = W1 (x - a k ) ] . For x -ft , any factor of 9 k , Wk(x - ft) - min{Yk, W 0(a k - ft )) , and W'(x - ft) > min{y k, W 0(a k - ft )} . The inequality cannot hold; for, then, W(x - ft ) > Yk - Wk(x - a k) > Wk(x - ft ) > \ ^ ( x - ft ) , which contradicts the choice of x - a k . Therefore W» (x - /3) m wk(x - ft) for a l l factors of ^ ; so l T 9 k = W k 9 k . Since 9k(<*k) • 0 , certainly x - a k | 9 k i n Wk . 26. Jhis means the redaction w i l l use only keys y± for i < k . But Wk9k = Wfcpk = pk > Wk-l9k ,hence the reduction of must be . Theorem 5.2: If a value W» = [ w o , W» (x - ft) = s] on ACxl reduces to V k • £ v o , V^x • M-I» ••• » vk^k ** ̂ k"] o n R C x l , then there exists an a , such that <pfc(a) - 0 , and a Y so that W* = W - [ w 0 , W(x - a) «Y] . Also Y = 8 . Proof: Let V k be extended to W on A[xV as in theorem 5*1} where a » a k , then W < Wf . If there exists an x - © e A[x] such that W(x - 9) < W»(x - Q) , then for G(x) e R[x] , where G (©) = 0, i t follows that VkG(x) = WG(x) < ¥»G(x) = VkG(x)=; hence, W = W . Also, since Y • W(x -<x)=W f(x-a)<8 and 5 = W»(x - ft ) - W(x - ft) <Y, then Y " 8 . Theorem 5.3: Let W« = [ w o , W^x - ft\) = S i , ... , W*(x-/^-Sj,. be a value on A £x] which reduces to v s [vo» v l x a ^1* ••• » V k 8 8 ̂ k> •••) o n R t x l > then W* may be represented by W = £Wq, Wjx = W2(x - a 2) « Y 2 , ... , Wk(x - a k) • Yk .. where: (1) 9 k(a k) = 0 f or a l l k > 1 , (2) reduction of Wk to R fx] is Vfc , Proof: The proof i s similar to that for 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 that each a k i s a root of the J27. -corresponding key cp̂  appearing i n the reduction of the valua- tion to R [ x ] • This information indicates how a valuation V of Rtx] may be extended to some valuation of A fx] . It wi l l now be shown how this extension can be accomplished. Lemma 5.4: Let a be a root of some polynomial 9 e Atx] , then a valuation W • Jw o , W(x - a) • If] can be defined on A[x_ such that 9 has a prescribed value M> • The value J* i s uniquely determined* n n—1 Proof: Let 9 = ft n ( x - a) + ft - a) + ... + ft^ix - a) , where ft± B k t then W9 = min | W0 ft ±+ i^}« Let the numbers Y± be defined by WQfti + lY^ • jx for i = 1 , 2 , . . . . t m and defined by Y - max JTi . For this value of Jf, i WQ / ? i + i J* > jx for i • 1 , 2 , ... , n and the equality holds for at least one value of i • Suppose there were two values Y and T , with Y > J*' > with the desired property. Since 9(a) = G , there exists an i ^ 0 such that I* - Wo ^ 1 + i y > WoA + 1 **' • So W defined by Y 1 would give 9 a value W9 < \i . There- fore the value W(x - a) i s unique. Lemma 5.5: In an inductive value Vk = [ Vo» V l x 1 3 "1> ••• » V l ^ k - l " *k-tfk*k - »k\ o f R W the for 1 < i < k are the complete, and only, set of ap- proximant s to 9 k • Proof: This follows immediately from [3] , theorem 5.3 (Cf • end of §3 ) i f V is defined by V 88 fVo» V l x s — » v k - l * k - l = h c - l ^ k s .Theorem 5.6: Let WQ on A be an extension of V Q on R , then V k * [v0, V^x = Hi, • •• » Vfc9k * o n R C x l may be extended to W * £w0, W(x - a) on A[ X] , where 9k(<*) = 0* Proof: By lemma 5.4 the value W(x - a) = Y i s uniquely deter- mined from Wcpk = Hk • Now, W defined by this reduces to y/ = f vo» v i x - ^i» » s *tr-i» * v ] > where J < k , by theorem 4.7 since 9 k(a). ? 0 • But lAtyk = Hk and, so, V̂ 1 9 k = M„ 88 ^k^k • This implies yf • k and v » Hk • That i s , the reduction of W is \ar W i s the extension of Vk • Since theorem 5.2 elaims every valuation W may be defined by some a where 9k (a) • 0 , i t i s seen that for a given W0 the maximum number of extensions of V k on R Tx] to W on A Cxi i s the degree of <pk 5 and, every extension may be found by the method of theorem 5.6 • Theorem 5.7: Let WQ on A be an extension of V Q on R and let V •"£-V0, Vpc * H I , ... , Vk9k 88 H-k» •••] be a limit value of Rrx] . The value V may be extended to the MacLane value W = ptf0, Wjx = HI, ¥ 2(x - a 2) = Y2> ••• » w k ^ x " ak^ = *̂k» ••• on ACx] where: (1) ^(c^) = 0 for i = 1, 2, ... , (2) i s uniquely determined by the factor x - a i and the value Hi of 9i , (3) P^(x - a±) =» 1 in x - a u l for a l l i > 1 (Cf. definition 3.2) , (4) Wi reduces to Vi on R[x] for i - 1, 2, ... . Proof: It w i l l be possible to formally construct the sequence of values {Wi} i f i t can be. shown that there always exists a factor 29. of 9̂  satisfying (3) . The construction of with this factor maybe accomplished by the method of theorem 5.6. How- ever, i t w i l l be necessary to show1 that W defined by this sequence of values is actually a MacLane value. In order to prove that property (3) may be satisfied i t w i l l be shown that, given a value WK which satisfies (1), (2), and (4) a value W k + i can be defined over WK also satisfying ( l ) , (2) and (4) and such that Dq,(x - a k + i ) =1 i n x - a k • Then, since satisfies (1), (2) and (4) i t follows by induction that property (3) can • be satisfied for a l l i > 1 . L e t WK satisfy (1), (2) and (4) and let W£ be defined by WFC = [W Q , WJ(x - a k) - tf] with Y < yk but such that Vk-1<pk < W q̂>k * y, < n k . Then as y "* » M- "* l̂ k • Therefore, i f <pk+i i s expanded i n powers of <pk , as ¥-*- &*k the value of 9 k + i must continuously i n - crease, since proj V k > 0 with respect to <Pk+l* So, i f 9k+l i s expanded i n powers of x - a k , the value of <pk+i must also con- tinuously increase as Y "*Tk • This could only happen i f l ^ k + l 2: 1 i n x - a k . Hence, there exists a factor x - a k + 1 °^ 9k+l s u c n *hat D^(x - a k + ^ ) • 1 i n x - a k . Now W k + i may be defined by W K + 1 = [ w k > W k + i ( x - <*k+1) * ^k+l] where W K +^ satisfies O D , (2), (3) and (4) . It only remains to show that W defined by this sequence of values is a MacLane value. Since D 9(x - a k + 1 ) =1 i n x - a k , Y k - ¥ k(x - a k) = W k(x - a k + 1 ) < W 0 ( a k - a k +^) ; that i s , J* k < W Q(a k " ak+l^ • ^et wk+l D e defined by wk+l - [ wo» w k f l ( x - a k + 1 ) = yk] , then WK = W K + 1 ; for, let x - ft e ACx] , then 30. W k( x - ft ) = min { Ykf W 0[a k - ft)] and w k + l > " ft > - m i n { Y k , W 0(a k + 1 - ) } but V< ak +1 " ft ) ̂  min { w 0 ( a k + 1 - a k ) , W Q(a k - /3 )} > min { f k , W0(ak-/3)] Therefore Ŵ  and w k + ^ have the same reduction on Rrx] • Hence, > i n order that W k +i reduce to V k +^ . Let Let <pk have the expansion <pk * f n ^ x " a k ^ n + f n - l ^ x ~ ak^ n"" 1 + ••• + f^(x - a k) ,then W ' m * n o { V i + 1 *\} a n d w k + i * k * j V i + i W k + i ( x - a k } l • Let i , which cannot be zero, be chosen such that a minimum term i s actually obtained in the second inequality; then Wf + i i * > W © • W <D > W f +iW (Y . a ) -0*1 1 'k - V k k+l 9k ~ o 1! 1 k + l v x °V * Therefore, since 1 ^ 0 , fk > Wk+1<x;:- a k) . Now, w o ( a k " ak+l J - B i n / w k + l ( x " ak+l>> w k + l ( x - ak>} » b u t W k + 1(x - a k + 1 ) = y k+l > **k £ w k + l ( x ~ ak>> a n d> s o » w o K - ak+l> • w k + l ( x " ak> ̂  **k • But i t i s known from above that W0(ak - otk+i) £ ^ . Therefore W Q(a k - a k + i ) • Y k and since $k+i > Yk for a l l k t 1 the sequence {ai} i s pseudo-convergent. The sequence has no pseudo- limit since W reduces to a limit value; otherwise V would be an inductive value ({2} , theorem 16.4) • There can be no limit for the sequence {aj.} i n A since V is a f i n i t e value; alter- natively, every limit of pseudo-convergent sequence i s a pseudo- li m i t . Therefore, W i s a MacLane valuation of A [x] satisfying properties (1), (2), (3) and Because of theorem 5»3 and since for every limit value W of Afx], D<p(x - a-k+i) = 1 i n x - a k , every extension of a limit value V of R[xl to A Cxi may be found as i n Theorem 5.7. Bibliography 1. A. Ostrowski, Untersuchen zur arithmetischen Theorie der KBrper, Mathematische Zeitschrift, vol. 39 (1934) , pp. 269 - 404. 2. S. MacLane. A construction for Absolute Values i n Polynomial Rings, Transactions of the American Mathematical Society, vol. 40 (1936) , pp. 363 - 395. 3 . S. MacLane, A construction for Prime Ideals as Absolute Values of an Algebraic Field, Duke Mathematical Journal, vol. 2 (1936) , pp. 492-510.

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