07-401/Class Notes for April 11: Difference between revisions
No edit summary |
|||
| Line 49: | Line 49: | ||
===Proof of The Fundamental Theorem=== |
===Proof of The Fundamental Theorem=== |
||
'''Proof of <math>\Psi\circ\Phi=I</math>.''' More precisely, we need to show that if <math>K</math> is an intermediate field between <math>E</math> and <math>F</math>, then <math>E_{\operatorname{Gal}(E/K)}=K</math>. The inclusion <math>E_{\operatorname{Gal}(E/K)}\supset K</math> is obvious, so we turn to prove the other inclusion. |
|||
'''Proof of <math>\Psi\circ\Phi=I</math>.''' |
|||
'''Proof of <math>\Phi\circ\Psi=I</math>.''' |
'''Proof of <math>\Phi\circ\Psi=I</math>.''' |
||
Revision as of 14:43, 9 April 2007
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 [math]\displaystyle{ F }[/math] will be a field of characteristic 0.
Statement
Theorem. Let [math]\displaystyle{ E }[/math] be a splitting field over [math]\displaystyle{ F }[/math]. Then there is a correspondence between the set [math]\displaystyle{ \{K:E/K/F\} }[/math] of intermediate field extensions [math]\displaystyle{ K }[/math] lying between [math]\displaystyle{ F }[/math] and [math]\displaystyle{ E }[/math] and the set [math]\displaystyle{ \{H:H\lt \operatorname{Gal}(E/F)\} }[/math] of subgroups [math]\displaystyle{ H }[/math] of the Galois group [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] of the original extension [math]\displaystyle{ E/F }[/math]:
The bijection is given by mapping every intermediate extension [math]\displaystyle{ K }[/math] to the subgroup [math]\displaystyle{ \operatorname{Gal}(E/K) }[/math] of elements in [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] that preserve [math]\displaystyle{ K }[/math],
and reversely, by mapping every subgroup [math]\displaystyle{ H }[/math] of [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] to its fixed field [math]\displaystyle{ E_H }[/math]:
Furthermore, this correspondence has the following further properties:
- It is inclusion-reversing: if [math]\displaystyle{ H_1\subset H_2 }[/math] then [math]\displaystyle{ E_{H_1}\supset E_{H_2} }[/math] and if [math]\displaystyle{ K_1\subset K_2 }[/math] then [math]\displaystyle{ \operatorname{Gal}(E/K_1)\gt \operatorname{Gal}(E/K_1) }[/math].
- It is degree/index respecting: [math]\displaystyle{ [E:K]=|\operatorname{Gal}(E/K)| }[/math] and [math]\displaystyle{ [K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)] }[/math].
- Splitting fields correspond to normal subgroups: If [math]\displaystyle{ K }[/math] in [math]\displaystyle{ E/K/F }[/math] is a splitting field then [math]\displaystyle{ \operatorname{Gal}(E/K) }[/math] is normal in [math]\displaystyle{ \operatorname{Gal}(E/F) }[/math] and [math]\displaystyle{ \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 1. Let [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] be algebraic elements of some extension [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math]. Then there exists a single element [math]\displaystyle{ c }[/math] of [math]\displaystyle{ E }[/math] so that [math]\displaystyle{ F(a,b)=F(c) }[/math]. (And so by induction, every finite extension of [math]\displaystyle{ 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. [math]\displaystyle{ \Box }[/math]
Splitting Fields are Good at Splitting
Lemma 2. (Compare with Hungerford's Theorem 10.15 on page 355). If [math]\displaystyle{ E }[/math] is a splitting field over [math]\displaystyle{ F }[/math] and some irreducible polynomial [math]\displaystyle{ p\in F[x] }[/math] has a root [math]\displaystyle{ v }[/math] in [math]\displaystyle{ E }[/math], then [math]\displaystyle{ p }[/math] splits in [math]\displaystyle{ E }[/math].
Proof. Let [math]\displaystyle{ L }[/math] be a splitting field of [math]\displaystyle{ p }[/math] over [math]\displaystyle{ E }[/math]. We need to show that if [math]\displaystyle{ w }[/math] is a root of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ L }[/math], then [math]\displaystyle{ w\in E }[/math] (so all the roots of [math]\displaystyle{ p }[/math] are in [math]\displaystyle{ E }[/math] and hence [math]\displaystyle{ p }[/math] splits in [math]\displaystyle{ E }[/math]). Consider the two extensions
The "smaller fields" [math]\displaystyle{ F(v) }[/math] and [math]\displaystyle{ F(w) }[/math] in these two extensions are isomorphic as they both arise by adding a root of the same irreducible polynomial ([math]\displaystyle{ p }[/math]) to the base field [math]\displaystyle{ F }[/math]. The "larger fields" [math]\displaystyle{ E=E(v) }[/math] and [math]\displaystyle{ E(w) }[/math] in these two extensions are both the splitting fields of the same polynomial ([math]\displaystyle{ f }[/math]) over the respective "small fields", as [math]\displaystyle{ E/F }[/math] is a splitting extension for [math]\displaystyle{ f }[/math] and we can use the sub-lemma below. Thus by the uniqueness of splitting extensions, the isomorphism between [math]\displaystyle{ F(v) }[/math] and [math]\displaystyle{ F(w) }[/math] extends to an isomorphism between [math]\displaystyle{ E=E(v) }[/math] and [math]\displaystyle{ E(w) }[/math], and in particular these two fields are isomorphic and so [math]\displaystyle{ [E:F]=[E(v):F]=[E(w):F] }[/math]. Since all the degrees involved are finite it follows from the last equality and from [math]\displaystyle{ [E(w):F]=[E(w):E][E:F] }[/math] that [math]\displaystyle{ [E(w):E]=1 }[/math] and therefore [math]\displaystyle{ E(w)=E }[/math]. Therefore [math]\displaystyle{ w\in E }[/math]. [math]\displaystyle{ \Box }[/math]
Sub-lemma. If [math]\displaystyle{ E/F }[/math] is a splitting extension of some polynomial [math]\displaystyle{ f\in F[x] }[/math] and [math]\displaystyle{ z }[/math] is an element of some larger extension [math]\displaystyle{ L }[/math] of [math]\displaystyle{ E }[/math], then [math]\displaystyle{ E(z)/F(z) }[/math] is also a splitting extension of [math]\displaystyle{ f }[/math].
Proof. Let [math]\displaystyle{ u_1,\ldots u_n }[/math] be all the roots of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ E }[/math]. Then they remain roots of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ E(z) }[/math], and since [math]\displaystyle{ f }[/math] completely splits already in [math]\displaystyle{ E }[/math], these are all the roots of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ E(z) }[/math]. So
and [math]\displaystyle{ E(z) }[/math] is obtained by adding all the roots of [math]\displaystyle{ f }[/math] to [math]\displaystyle{ F(z) }[/math]. [math]\displaystyle{ \Box }[/math]
Proof of The Fundamental Theorem
Proof of [math]\displaystyle{ \Psi\circ\Phi=I }[/math]. More precisely, we need to show that if [math]\displaystyle{ K }[/math] is an intermediate field between [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math], then [math]\displaystyle{ E_{\operatorname{Gal}(E/K)}=K }[/math]. The inclusion [math]\displaystyle{ E_{\operatorname{Gal}(E/K)}\supset K }[/math] is obvious, so we turn to prove the other inclusion.
Proof of [math]\displaystyle{ \Phi\circ\Psi=I }[/math].
Proof of Property 1.
Proof of Property 2.
Proof of Property 3.
