現代数学の系譜工学物理雑談古典ガロア理論も読む78

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む78 (1002ﾚｽ)
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466: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/04(月) 18:03:34.00 ID:Qu1TcOyQ

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”PARAMETRIC GALOIS EXTENSIONS”
https://arxiv.org/pdf/1310.6682.pdf
PARAMETRIC GALOIS EXTENSIONS
FRANC， OIS LEGRAND
Abstract. Given a field k and a finite group H, an H-parametric
extension over k is a finite Galois extension of k(T ) of Galois group
containing H which is regular over k and has all the Galois extensions of k of group H among its specializations. We are mainly interested in producing non H-parametric extensions, which relates
to classical questions in inverse Galois theory like the BeckmannBlack problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper
and provide new non parametricity criteria and new examples.

1. Presentation
The Inverse Galois Problem asks whether, for a given finite group H,
there exists at least one Galois extension of Q of group H. A classical
way to obtain such an extension consists in producing a Galois extension E/Q(T) with the same group which is regular over Q 1
: from the Hilbert irreducibility theorem, E/Q(T) has at least one specialization
of group H (in fact infinitely many if H is not trivial).
In this paper we are interested in “parametric Galois extensions”,
i.e. in finite Galois extensions E/Q(T) which are regular over Q - from
now on, say for short that E/Q(T) is a “Q-regular Galois extension”
- and which have all the Galois extensions of Q of group H among
their specializations. More precisely, given a field k and a finite group
H, we say that a k-regular finite Galois extension E/k(T) of group G
containing H (with possibly H 6= G) is H-parametric over k if any
Galois extension of k of group H ocurs as a specialization of E/k(T)
(definition 2.2). The special case H = G is of particular interest.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/466

467: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/04(月) 18:03:59.86 ID:Qu1TcOyQ

>>466
つづき

This was introduced in our previous paper [Leg13b] in the number
field case. Given a field k and a finite group G, the question of whether
there is a G-parametric extension over k of group G or not is intermediate between these classical two questions in inverse Galois theory:
- if there is such an extension, then it obviously solves the BeckmannBlack problem for G over k, which asks whether any Galois extension
F/k of group G occurs as a specialization of some k-regular Galois
extension EF /k(T) with the same group,
- if there are no such extension, then there obviously cannot exist a
one parameter generic polynomial over k of group G, i.e. a polynomial
P(T, Y ) ∈ k(T)[Y ] of group G such that the splitting extension over
L(T) is G-parametric over L for any field extension L/k.
We refer to §2.2 for more details.
If studying parametric extensions indeed seems a natural first step to
these important topics, it is itself already quite challenging, especially
over number fields. The question of deciding whether a given k-regular
Galois extension of k(T) of given group G is G-parametric over a given
base field k or not indeed seems to be difficult, even for small groups G:
for example, in the case G = Z/3Z and k = Q, the answer seems to be
known for only one such extension (this extension is Z/3Z-parametric
over Q; see §1.1 below).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/467

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