Field Of Formal Power Series Research Articles

Manin homomorphisms for elliptic curves over a field of formal power series

In this paper we describe the intersection of the kernels of Manin homomorphisms for an arbitrary elliptic curve over a field of formal power series.

On linear representations of a matrix group with elements from a field of formal power series over an infinite field

On linear representations of a matrix group with elements from a field of formal power series over an infinite field

An algebraically closed field

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

Transference theorems in the field of formal power series

Transference theorems in the field of formal power series

Analogue of a theorem of Khintchine in fields of formal power series

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series

The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series

On the existence of real-closed fields that are $\eta \sb{\alpha }$-sets of power $\aleph \sb{\alpha }$

Introduction. In the theory of 7la-sets three main theorems stand out: I. An 77-set is universal for totally ordered sets of power not exceeding Ka. II. Two 71a-sets of power R. are isomorphic. III. If R. is regular and if Ea 0 then these three theorems hold for the category of totally ordered Abelian groups and order preserving (group) isomorphisms; the special group being totally ordered, Abelian, divisible, and an 77a-set. Erdos, Gillman, and Henriksen [8] (see also Gillman and Jerison [11]) proved that if a> 0 then I and II hold for the category of totally ordered fields and order preserving (ring) isomorphisms; the special field being real-closed and an 7a-set. It was also shown in [8] that III holds for this category and special object if a = 1. However in case a > 1, III was left open both in [8] and in [11]. The initial aim of these researches was to show that, assuming a> 0, K, regular, and Ea 0. Let R { G } denote the field of formal power series with exponents in G and coefficients in R, the reals. R{G } is an 7la-set but its power exceeds K. Let R { G } a = {f ER { G }: the support of f is of power less than a }. Then R{G } a, again a real-closed field, is an 77a-set, and is of power K,. The only difficult point in these verifications was the proof that R { G } and R{G}a are 77a-sets. The proof arrived at by the author did not involve the multiplication in these fields, but depended wholly on their structure as a

Division Algebras Over Fields of Formal Power Series

Division Algebras Over Fields of Formal Power Series

Division algebras over fields of formal power series

and select p>m, p>n. Let C=AB+N, D=BA. Then L=LCP 09(CP), M=MDPEfD%(DP). We shall prove that A induces an isomorphism A' on LCp onto MDP. Since for each j, CiA =ADi, we have LCPA =LADPCMDP. But MDP=MDP+1=MB(AB)PA =MBCPA CLCPA. Thus A is on LCP onto MDP. We observe that for any r, Cr=(AB)r+(AB)r-lN+ * * * +Nr, hence for r=2p, C2P=(AB)2P + (AB) 2p-'N+ * * * + (AB)P+'NP-1 since NP =Nm = o. If xCPA = 0, then x(AB)PA =O, x(AB)P+1=x(AB)p+2 = * =0. Thus XC2p=O, xCp=O, which proves that A' is an isomorphism. We finally have CA'=A'D so that the contraction of C to LCP is similar to the contraction of D to MDP. Thus C and D have the same elementary divisors which do not have zero as a root. The theorem follows from Theorem 2.

Automorphisms of fields of formal power series

1. S. Ramanujan, Congruence properties of partitions. Math. Zeit. vol. 9 (1921) pp. 147-153. Reprinted in Collected papers of Srinivasa Ramanujan, Cambridge, 1927, pp. 232-238. 2. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. vol. 22 (1916) pp. 159-184. Reprinted in Collected papers of Srinivasa Ramanujan, pp. 136-162. 3. H. S. Zuckerman, Identities analogous to Ramanujan's identities involving the partition f unction, Duke Math. J. vol. 5 (1939) p. 89, equation (1.15).

The universality of formal power series fields

The universality of formal power series fields

Arithmetic in Fields of Formal Power Series in Several Variables

Arithmetic in Fields of Formal Power Series in Several Variables