Units and class groups in cyclotomic function fields

Galovich, Steven; Rosen, Michael

doi:10.1016/0022-314x(82)90045-2

Cited by 43 publications

(24 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Let j be a generator of J as before, so Tables 1 and 2. Let P be an infinite prime of K and p be the infinite prime of K + lying below P. From [GR,Proposition 1.10], there exists a primitive M -torsion point λ such that ord P (λ) = (d − 1)(q − 1) − 1, where d = deg M . As p is totally ramified in K, we also have ord p (λ q−1 ) = (d − 1)(q − 1) − 1.…”

Section: First We Make a Partition Of The Column Indicesmentioning

confidence: 99%

On the relative class number of cyclotomic function fields

Jung

Ahn

2003

Acta Arith.

View full text Add to dashboard Cite

Section: First We Make a Partition Of The Column Indicesmentioning

confidence: 99%

On the relative class number of cyclotomic function fields

Jung

Ahn

2003

Acta Arith.

View full text Add to dashboard Cite

“…The action u »-» uM gives the additive group of F9(F)ac the structure of an F/--module. The following properties hold: (1) If the degree of Af is d, then A^ = {X \ XM = 0} contains qd elements. Moreover, AM is a cyclic Fj-module, isomorphic to Rj/{M), for every Af 0 in Fr.…”

Section: Subfields Of Cyclotomic Function Fieldsmentioning

confidence: 99%

“…The ordinary distribution that we will concentrate on was constructed by Galovich-Rosen [1]. Let x = A/N e k/RT, where A, N e RF and deg(^f) = where the product is taken over the monic prime polynomials which divide F .…”

Section: Distributionsmentioning

confidence: 99%

Circular Units of Function Fields

Harrop¹

1994

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Abstract.A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.Let m be a positive integer and let C,m = exp{2ni/m).Let Km -Q{Çm) denote the mux cyclotomic field, and F+ its maximal real subfield. There is a parallel setup for function fields of characteristic p. Let ¥q be the finite field with q elements, let RF = ¥q[T] be the ring of polynomials over ¥q (with F transcendental over Fi), and let ¥q{T) be the field of rational functions over ¥q . To each polynomial M e RF one can associate an extension KM, called the Afth cyclotomic function field, which enjoys properties analogous to those of the cyclotomic number field Km . In particular, Galovich and Rosen [1] proved the analogue of Sinnott's theorem in this setting. The purpose of this paper is to extend this unit index-class number formula to an arbitrary subfield of KM.Let k be any subfield of the Afth cyclotomic function field (Af monic), G the Galois group of k over ¥q{T), k+ the maximal subfield of k in which oo splits, Ok{Ok+) the integral closure of ¥q [T] in k{k+), and 0*k the unit group of Ok . In §3, we define a subgroup C of Ok , which we call the circular units of k. Our main result is that C has finite index in Ok , and that this index may be written in the form [Ok':C] = h(Ok+)'4, where h{Ok+) is the class number of Ok+, and ck is a rational number whose definition does not involve h{Ok+).We now briefly describe the contents of the rest of this paper. In §1, we present the relevant definitions and facts in the function field setting. We also state the analytic class number formula. In §2, we review ordinary distributions on ¥q{T)/Rr, discuss an index notation, and obtain a preliminary result on the structure of a certain module. The circular units are introduced in §3 and

show abstract

“…Let A be a monic irreducible polynomial. Suppose that K = k(Λ A ) is the Ath cyclotomic function field and K + is the maximal real subfield of K. In Section 5, we will prove that such places exist for K and K + if the class number of O K + is relatively prime to d. It should be noted that the proof heavily relies on Galovich and Rosen's work on Sinnott's circular units in cyclotomic function fields [3].…”

mentioning

confidence: 99%

“…In this case, Galovich and Rosen proved (see [3]) For each polynomial W relatively prime to A there is a unique element σ W ∈ G such that σ W (λ) = ρ W (λ) where λ is a primitive A-torsion element (see Theorem 12.8 of [7]). Using the definition of the group ring action, the multiplicity of σ N , and cancellation in a telescoping product, we have…”

mentioning

confidence: 99%

A note on global units and local units of function fields

2009

Acta Arith.

View full text Add to dashboard Cite

Units and class groups in cyclotomic function fields

Cited by 43 publications

References 10 publications

On the relative class number of cyclotomic function fields

On the relative class number of cyclotomic function fields

Circular Units of Function Fields

A note on global units and local units of function fields

Contact Info

Product

Resources

About