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28 August 2011
- 20:1620:16, 28 August 2011 diff hist +1,271 N Galois Group is Group Created page with "== Theorem == Let $L/K$ be a normal extension and :$\operatorname{Gal}(L/K) = \{\sigma:L\mapsto L \mid \sigma$ is an [[Definition:Automorphism#..."
26 August 2011
- 04:5304:53, 26 August 2011 diff hist −477 Minimal Polynomial is Unique →Proof
- 04:3804:38, 26 August 2011 diff hist +923 N Definition:Galois Group of Field Extension Created page with "== Definition == Let $L/K$ be a Galois extension. Then the set :$\operatorname{Gal}(L/K) = \{\sigma:L\mapsto L$ | $\sigma$ is an [[Definition:Au..."
- 04:2204:22, 26 August 2011 diff hist +54 Fundamental Theorem of Galois Theory →Theorem
- 04:2004:20, 26 August 2011 diff hist +2,388 Fundamental Theorem of Galois Theory →Proof
25 August 2011
- 01:3801:38, 25 August 2011 diff hist +27 Abstract Model of Algebraic Extensions →Proof
- 01:3301:33, 25 August 2011 diff hist +28 Order of Galois Group Equals Degree of Extension →Proof
- 01:3201:32, 25 August 2011 diff hist +1,434 N Order of Galois Group Equals Degree of Extension Created page with "== Theorem == Let $L/K$ be a Galois extension. Then $|\operatorname{Gal}(L/K)| = [L : K]$. == Proof == Since $L/K$ is [[Definition:Galois Extens..."
- 00:5200:52, 25 August 2011 diff hist +374 N Definition:Inclusion-Reversing Mapping Created page with "== Definition == Let $\phi:A\mapsto B$ be a function and $A,B$ be sets of sets. Then $\phi$ is '''inclusion-reversi..."
24 August 2011
- 06:2506:25, 24 August 2011 diff hist +311 N First Fundamental Theorem on Ring Homomorphisms Created page with "== Theorem == Let $R$ and $S$ be rings and $\phi:R\mapsto S$ be a ring homomorphism. Then :$S\cong R/\operatorn..."
- 06:1106:11, 24 August 2011 diff hist −1 Equivalence of Definitions of Normal Extension →Proof
- 06:1006:10, 24 August 2011 diff hist +871 N Abstract Model of Algebraic Extensions Created page with "== Theorem == Let $K$ be a field and $\alpha \in \overline{K}$ be an element of the algebraic closure of ..."
- 05:4805:48, 24 August 2011 diff hist +361 User talk:Prime.mover →Notation
- 05:4205:42, 24 August 2011 diff hist −6 Schönemann-Eisenstein Theorem →Theorem
- 05:4205:42, 24 August 2011 diff hist −7 Schönemann-Eisenstein Theorem →Theorem
- 03:5303:53, 24 August 2011 diff hist −1 Definition:Embedding (Galois Theory) →Definition
- 03:5203:52, 24 August 2011 diff hist +208 N Definition:Embedding (Galois Theory) Created page with "== Definition == Let $K$ and $L$ be fields. A monomorphism $\phi:K\mapsto L$ is called an ..."
- 03:4703:47, 24 August 2011 diff hist +1,631 Equivalence of Definitions of Normal Extension →Theorem
- 03:2803:28, 24 August 2011 diff hist +25 Fundamental Theorem of Galois Theory →Theorem
- 03:2803:28, 24 August 2011 diff hist +183 Fundamental Theorem of Galois Theory →Proof
- 03:2303:23, 24 August 2011 diff hist +34 Fundamental Theorem of Galois Theory →Theorem
- 03:0603:06, 24 August 2011 diff hist +461 N Equivalence of Definitions of Normal Extension Created page with "== Theorem == Let $L/K$ be a field extension. Then the following are equivalent: #For every [[Definition:Irreducible (Ring Theory)#Polynomials|irre..."
- 02:5802:58, 24 August 2011 diff hist +53 Definition:Normal Extension →Definition
- 02:5702:57, 24 August 2011 diff hist +40 Definition:Normal Extension →Alternative Definition
- 02:5602:56, 24 August 2011 diff hist +182 Fundamental Theorem of Galois Theory →Theorem
- 02:5602:56, 24 August 2011 diff hist +200 N Definition:Intermediate Field Created page with "== Definition == Let $L/K$ be a field extension. Then a field $F$ is called an '''intermediate field''' if ..."
- 02:4702:47, 24 August 2011 diff hist −15 User talk:Prime.mover →Notation
- 02:4502:45, 24 August 2011 diff hist +1,195 User talk:Prime.mover →Notation: new section
23 August 2011
- 19:0419:04, 23 August 2011 diff hist +946 N Minimal Polynomial is Irreducible Created page with "== Theorem == Let $L/K$ be a field extension and $\alpha\in L$ be algebraic over $K$. Then the [[Definition:Minimal Polyn..."
- 18:5218:52, 23 August 2011 diff hist +200 N Definition:Galois Extension/Finite Created page with "== Definition == A finite degree field extension $L/K$ is '''Galois''' if it is normal and [[Definition:Separable ..."
- 18:4918:49, 23 August 2011 diff hist +26 Minimal Polynomial is Unique →Proof
- 18:4718:47, 23 August 2011 diff hist +1,102 N Minimal Polynomial is Unique Created page with "== Theorem == Let $L/K$ be a field extension and $\alpha\in L$ be algebraic over $K$. Then the [[Definition:Minimal Polyn..."
- 18:3118:31, 23 August 2011 diff hist +216 N Definition:Separable Extension Created page with "== Definition == A field extension $L/K$ is '''separable''' if for every $\alpha\in L$, the minimal polynomial of..."
- 18:2718:27, 23 August 2011 diff hist 0 Definition:Normal Extension →Definition
- 18:2618:26, 23 August 2011 diff hist +1,017 N Definition:Normal Extension Created page with "== Definition == A field extension $L/K$ is a '''normal extension''' if for every irreducible polynomial $f\in K[x]$ with at least one root in $L$,..."
- 18:1518:15, 23 August 2011 diff hist +511 N Definition:Algebraic Closure Created page with "== Definition == Let $K$ be a field, then by the existence of algebraic closures there is some algebraically closed field $\overline{K}$ c..."
- 04:5204:52, 23 August 2011 diff hist +1,532 Fundamental Theorem of Galois Theory →Theorem
- 04:4404:44, 23 August 2011 diff hist +351 N Ring of Algebraic Integers Created page with "== Definition == Let $\mathbb{A}$ denote the set of all complex numbers which are the root of some monic polynomial in $\Z[x]$. It can be shown that $\mathbb{A}$ is a ring, cal..."
- 04:4404:44, 23 August 2011 diff hist +31 Integral Multiple of an Algebraic Number →Theorem
- 04:3204:32, 23 August 2011 diff hist −44 Schönemann-Eisenstein Theorem →Theorem
- 03:4103:41, 23 August 2011 diff hist +413 N Fundamental Theorem of Galois Theory Created page with "== Theorem == Let $L/K$ be a finite, Galois extension. Let $H$ denote a subgroup of $\text{Gal}(L/K)$ and $F$ denote an intermediate field. Then the functions $H \xrightarrow..."
19 August 2011
- 19:3019:30, 19 August 2011 diff hist +1,208 Gauss's Lemma on Primitive Rational Polynomials →Proof
- 19:0619:06, 19 August 2011 diff hist +6 Gauss's Lemma on Primitive Rational Polynomials →Proof
- 19:0619:06, 19 August 2011 diff hist +6 Gauss's Lemma on Primitive Rational Polynomials →Proof
- 19:0519:05, 19 August 2011 diff hist −2 Gauss's Lemma on Primitive Rational Polynomials →Theorem
- 19:0319:03, 19 August 2011 diff hist +451 Gauss's Lemma on Primitive Rational Polynomials →Proof
- 16:1616:16, 19 August 2011 diff hist −94 Schönemann-Eisenstein Theorem →Theorem
- 16:1416:14, 19 August 2011 diff hist +33 Schönemann-Eisenstein Theorem →Proof
- 16:1416:14, 19 August 2011 diff hist +845 N Gauss's Lemma on Primitive Rational Polynomials Created page with "== Theorem == If $f,g\in \Z[x]$ are primitive, then so is $h = fg.$ == Proof == First, we write $f = b_ex^e + b_{e-1}x^{e-1} + \ldots + b_0..."