192 Normat 59:3-4, 192 (2011)

Summary in English

Juliusz Brzezinski, What you should

know about cubic and quartic equations

The title says it all. This is intended

to be a standard reference for giving

the Galois group of cubic and especially

quartic polynomials.

Juliusz Brzezinski and Ulf Pers-

son, How to describe all cubic Galois

extensions?

Given a cubic Galois extension over Q

an algorithm for ﬁnding a minimal de-

ﬁning cubic is described in terms of Ei-

senstein integers. In particular one is ab-

le to determine whether two such cubic

extensions are isomorphic or not.

Christian U. Jensen, The inverse

problem of Galois theory

A survey of the inverse Galois problem

and variations thereof. Hilbert’s irredu-

cibility Theorem, the Noether problem

and the notion of G-generic polynomials

(G referring to a speciﬁc Galois group)

are discussed, both as approaches and

as elaborations. The most common Ga-

lois group over the rationals for polyno-

mials of degree n is S

n

but it is hard

to exhibit explicit families of polyno-

mials of degree n, n running through the

natural numbers, with S

n

as a Galois

group. It has been shown that all solvab-

le groups occur, and that all sporadic

simple groups with the possible excep-

tion of the Mathieu group M

23

occur as

well as many classical groups.

Juliusz Brzezinski, Galois groups and

number theory.

This provides a crash-course on alge-

braic number theory with special

empha- sizes of the role played by the

Galois groups. The standard facts on

splitting and ramiﬁcation of primes in

the rings of algebraic integers are pre-

sented, preparing for a short discus-

sion of class ﬁeld theory. Special at-

tention is given to the fact that eve-

ry abelian extension is contained in so-

me cyclotomic extension, the so cal-

led Kronecker-Weber theorem. Possib-

le generalizations of this theorem have

constituted a guiding inspiration for al-

gebraic numb er theory starting with

Kronecker himself (his famous Jugend-

traum) and included by Hilbert in his

famous list of twenty-three. For this

purpose the association of a holomorp-

hic function to an irreducible represen-

tation of the non-abelian Galois group,

the so called Artin L-function has be-

en ce ntral. Artin conjectured that those

should be entire functions and was ab-

le to verify it for 1-dimensional repre-

sentations via identiﬁcations with s uit-

able Dirichlet characters. This leads to

the ﬁrst taste of the Langlands pro-

gram and the possible connections with

automorphic c uspidal representations of

GL(n, Q). T his play a central role in

Wiles proof of Fermat’s Last Theorem,

and the connection with Taniyama-

Shimura-Weil and elliptic curves is sket-

ched, along with the deﬁnition of the

Hasse-Weil L-function. A brief discus-

sion on when the two L-functions can be

showed to be equal follows. The article

is capped by a few general comments on

the Langlands program.

Dennis Eriksson and Ulf Persson,

Galois theory and coverings

A connection between topological

covers and Galois theory is described,

with emphasis on the case of curves.

In particular one looks at holomorphic

maps f : C æ P

1

ramiﬁed at only three

points which are described by so called

dessins (d’enfants). It is shown how the

Galois group Gal(

Q/Q) acts on them.