07-401/Class Notes for April 11: Difference between revisions

From Drorbn
Jump to navigationJump to search
No edit summary
No edit summary
Line 2: Line 2:
{{In Preparation}}
{{In Preparation}}


The goal of today's class is to prove a (weak but strong enough) form of the '''Fundamental Theorem of Galois Theory''' as follows:
==The Fundamental Theorem of Galois Theory==


It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.
⚫
'''Theorem.''' Let <math>F</math> be a field of characteristic 0 and let <math>E</math> be a splitting field over <math>F</math>. Then there is a correspondence between the set <math>\{K:E/K/F\}</math> of intermediate field extensions <math>K</math> lying between <math>F</math> and <math>E</math> and the set <math>\{H:H<\operatorname{Gal}(E/F)\}</math> of subgroups <math>H</math> of the Galois group <math>\operatorname{Gal}(E/F)</math> of the original extension <math>E/F</math>:

Here and everywhere below our base field <math>F</math> will be a field of characteristic 0.

===Statement===

⚫
'''Theorem.''' Let <math>E</math> be a splitting field over <math>F</math>. Then there is a correspondence between the set <math>\{K:E/K/F\}</math> of intermediate field extensions <math>K</math> lying between <math>F</math> and <math>E</math> and the set <math>\{H:H<\operatorname{Gal}(E/F)\}</math> of subgroups <math>H</math> of the Galois group <math>\operatorname{Gal}(E/F)</math> of the original extension <math>E/F</math>:
{{Equation*|<math>\{K:E/K/F\}\quad\leftrightarrow\quad\{H:H<\operatorname{Gal}(E/F)\}</math>.}}
{{Equation*|<math>\{K:E/K/F\}\quad\leftrightarrow\quad\{H:H<\operatorname{Gal}(E/F)\}</math>.}}
The bijection is given by mapping every intermediate extension <math>K</math> to the subgroup <math>\operatorname{Gal}(E/K)</math> of elements in <math>\operatorname{Gal}(E/F)</math> that preserve <math>K</math>,
The bijection is given by mapping every intermediate extension <math>K</math> to the subgroup <math>\operatorname{Gal}(E/K)</math> of elements in <math>\operatorname{Gal}(E/F)</math> that preserve <math>K</math>,
Line 14: Line 20:
# It is degree/index respecting: <math>[E:K]=|\operatorname{Gal}(E/K)|</math> and <math>[K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]</math>.
# It is degree/index respecting: <math>[E:K]=|\operatorname{Gal}(E/K)|</math> and <math>[K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]</math>.
# Splitting fields correspond to normal subgroups: If <math>K</math> in <math>E/K/F</math> is a splitting field then <math>\operatorname{Gal}(E/K)</math> is normal in <math>\operatorname{Gal}(E/F)</math> and <math>\operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)</math>.
# Splitting fields correspond to normal subgroups: If <math>K</math> in <math>E/K/F</math> is a splitting field then <math>\operatorname{Gal}(E/K)</math> is normal in <math>\operatorname{Gal}(E/F)</math> and <math>\operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)</math>.

===Lemmas===

The two lemmas below belong to earlier chapters but we skipped them in class.

====The Primitive Element Theorem====

The celebrated "Primitive Element Theorem" is just a lemma for us:

'''Lemma.''' Let <math>a</math> and <math>b</math> be algebraic elements of some extension <math>E</math> of <math>F</math>. Then there exists a single element <math>c</math> of <math>E</math> so that <math>F(a,b)=F(c)</math>. (And so by induction, every finite extension of <math>E</math> is "simple", meaning, is generated by a single element, called "a primitive element" for that extension).

'''Proof.''' See the proof of Theorem 21.6 on page 375 of Gallian's book.

====Splitting Fields are Good at Splitting====

Revision as of 13:15, 9 April 2007

07-401/Navigation Panel  
# Week of... Links
1 Jan 10 About, Notes, HW1
2 Jan 17 HW2, Notes
3 Jan 24 HW3, Photo, Notes
4 Jan 31 HW4, Notes
5 Feb 7 HW5, Notes
6 Feb 14 On TT, Notes
R Feb 21 Reading week
7 Feb 28 Term Test
8 Mar 7 HW6, Notes
9 Mar 14 HW7, Notes
10 Mar 21 HW8, E8, Notes
11 Mar 28 HW9, Notes
12 Apr 4 HW10, Notes
13 Apr 11 Notes, PM
S Apr 16-20 Study Period
F Apr 24 Final
07-401 Class Photo.jpg
Add your name / see who's in!
Register of Good Deeds
In Preparation

The information below is preliminary and cannot be trusted! (v)

The Fundamental Theorem of Galois Theory

It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.

Here and everywhere below our base field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} will be a field of characteristic 0.

Statement

Theorem. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} be a splitting field over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . Then there is a correspondence between the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{K:E/K/F\}} of intermediate field extensions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} lying between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and and the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{H:H<\operatorname{Gal}(E/F)\}} of subgroups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} of the Galois group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/F)} of the original extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/F} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{K:E/K/F\}\quad\leftrightarrow\quad\{H:H<\operatorname{Gal}(E/F)\}} .

The bijection is given by mapping every intermediate extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} to the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/K)} of elements in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/F)} that preserve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K\mapsto\operatorname{Gal}(E/K)} ,

and reversely, by mapping every subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} of to its fixed field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_H} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\mapsto E_H} .

Furthermore, this correspondence has the following further properties:

  1. It is inclusion-reversing: if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1\subset H_2} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{H_1}\supset E_{H_2}} and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1\subset K_2} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/K_1)>\operatorname{Gal}(E/K_1)} .
  2. It is degree/index respecting: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [E:K]=|\operatorname{Gal}(E/K)|} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]} .
  3. Splitting fields correspond to normal subgroups: If in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/K/F} is a splitting field then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/K)} is normal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/F)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)} .

Lemmas

The two lemmas below belong to earlier chapters but we skipped them in class.

The Primitive Element Theorem

The celebrated "Primitive Element Theorem" is just a lemma for us:

Lemma. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} be algebraic elements of some extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . Then there exists a single element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a,b)=F(c)} . (And so by induction, every finite extension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is "simple", meaning, is generated by a single element, called "a primitive element" for that extension).

Proof. See the proof of Theorem 21.6 on page 375 of Gallian's book.

Splitting Fields are Good at Splitting

Retrieved from "https://drorbn.net/index.php?title=07-401/Class_Notes_for_April_11&oldid=4641"