ガロア第一論文及びその関連の資料スレ

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54: １３２人目の素数さん [sage] 2022/12/19(月) 08:24:07.78 ID:KRlSoN+A

下記について調べた
https://en.wikipedia.org/wiki/Galois_extension
Galois extension very irreducible polynomial in {\displaystyle F[x]}F[x] with at least one root in {\displaystyle E}E splits over {\displaystyle E}E and is separable.

(参考)
https://sites.pitt.edu/~gmc/
Gregory M. Constantine: Home Page
Professor of Mathematics
Department of Mathematics, The University of Pittsburgh

https://sites.pitt.edu/~gmc/
Greg Constantine: Fields
Final exam -- Autumn 2007
The course has four major themes:
1. The Tower Law, algebraic extensions, and constructions by straightedge and compass
2. The fundamental theorem of Galois theory.
3. The primitive element theorem and the fundamental theorem of algebra.
4. Insolvability by radicals of polynomial equations.

Resources used for Field Theory, are:
Kaplansky's "Fields and Rings", Chicago lectures in mathematics, The University of Chicago Press, 1972 [This is the main resource, by far.]
David Cox's "Galois Theory", Wiley, New York, 2004
Mark Dickinson's notes "Galois theory: The proofs, the whole proofs, and nothing but the proofs"

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/54

55: １３２人目の素数さん [sage] 2022/12/19(月) 08:24:45.57 ID:KRlSoN+A

>>54
つづき

・Galois theory notes （下記PDF）
・Field theory: homework sets

（下記PDFに冒頭の定理の証明がある）
https://sites.pitt.edu/~gmc/algebra/galoistheory.pdf
GALOIS THEORY: THE PROOFS, THE WHOLE PROOFS, AND
NOTHING BUT THE PROOFS
MARK DICKINSON
Contents
1. Notation and conventions 1
2. Field extensions 1
3. Algebraic extensions 4
4. Splitting fields 6
5. Normality 7
6. Separability 7
7. Galois extensions 8
8. Linear independence of characters 10
9. Fixed fields 13
10. The Fundamental Theorem 14

I’ve adopted a slightly different method of proof from the textbook for many of
the Galois theory results. For your reference, here’s a summary of the main results
and their proofs, without any of that pesky history and motivation?or distracting
examples?to get in the way. Just the proofs1
. Almost all of the hard work lies
in three main theorems: Corollary 7.9 (a splitting field of a separable polynomial
is Galois), Theorem 8.1 (linear independence of characters), and Theorem 9.1 (the
degree of K over KH is bounded by the order of H).

1 and the occasional definition or two. Not to mention the theorems, lemmas and so forth.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1615510393/55

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