(a) If p ramifies in Q(ζm), then p | m; the ramification index ep of the primes over p is φ(p) = (p− 1)pk−1 if p ‖ m. (b) The prime p is wildly ramified in Q(ζm) if and only if p | m. (c) If p m, then p is unramified. In the ring of integers Z[ζm] of Q(ζm), we obtain pOL = ∏gp i=1 pi. The primes pi have residue class degree fp = [Fp(ζm) : Fp], which (by looking at the Frobenius x 7→ x acting on Fp(ζm) = kpi) is equal to the order of p mod m in (Z/mZ)∗. The number of such primes is gp = #{pi|p} = [(Z/mZ)∗ : 〈p mod m〉]. In particular, a prime splits competely if and only if p ≡ 1 (mod m). If Q ⊂ L ⊂ L = Q(ζm) is a subfield given by a subgroup H ⊂ (Z/mZ)∗, then one has the Artin map

Class Field Theory

Algorithms and Programming

In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.

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Essential dimension: a survey

https://www.rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf

Class Field Towers

The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.

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Ramanujan's Forty Identities for the Rogers-ramanujan Functions

Parametrization of the Quadratic Fields Whose Class Numbers are Divisible by Three

Let k be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over k via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for \( \mathfrak{S} \)3 and C3 over k. As an application of the cubic case, we also give several sextic generic polynomials over k.

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

We study the field isomorphism problem for a cubic generic polynomial X 3 + sX + s via Tschirnhausen transformation. Through this process, there naturally appears a 2-dimensional involutive Cremona transformation. We show that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.

/pdf/tschirnhausen-transformation-of-a-cubic-generic-polynomial-1wicb5o32c.pdf

Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation

Let k be a eld of characteristic 2. We survey a general method of the eld intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.

Katsuya Miyake

Papers

Parametrization of the Quadratic Fields Whose Class Numbers are Divisible by Three

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation

On the field intersection problem of generic polynomials: a survey

Galois Theory and Modular Forms