"Differential Galois theory"의 두 판 사이의 차이

2009년 6월 29일 (월) 13:48 판

간단한 소개

There are two important conditions required in the Galois theory.

transitivity
fixed point free action

Sometimes, \(\text{Gal}(K/F)=|K:F|\) is presented as a condition. This automatically implies fixedpoint free transtive action.

How important is the transitivity in Galois theory and Monodromy theory?

regular covering

covering \(p:Y \to X\)

Deck transformation group =

\(\pi_1(X)\) is detecting the information between \(X\) and its universal covering.

If this is regular covering, then the group on the right side is same as the Galois group.

So we can understand the Galois group from homotopic theoretic viewpoint.

the role of algebraic closure

this is analogous to the universal covering.

Suppose we have field extensions \(\bar{F} , K, F\).

We associate some imaginary space \(X_F\) to the field \(F\).

And get a projection \(p:X_K \to X_F\)

Then we define the absolute Galois group \(\text{Gal}(\bar{F}/F)=\pi_1({X_F})\) as a homotopic concept.

\(\text{Gal}(K/F)=|K:F|\)

Gal(\bar{F}/F) = Gal(\bar{K}/K)

homotopy lifting theorem

Galois theory for prime ideals

dictionary

geometric viewpoints vs field theoretic viewpoints (or algebra vs. geometry)

covering space - field extensions

regular covering - Galois covering

degree of covering - degree of field extension

Spec of ring of integers - number fields

Riemann surfaces or algebraic curve - function field

example

\(x^3-2=0\)

\(K=\mathbb{Q}(\omega, \sqrt[3]{2})\) over \(\mathbb{Q}\)

\(Spec \mathbb{Z}[\omega,\sqrt[3]2]\) over \(Spec \mathbb{Z}\)

\([K : \mathbb{Q}]=6\)

Note that this degree is not equal to the degree of the polynomial.

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참고할만한 자료

\(\text{Gal}(K/F)=\pi_1(X_F)/p_{\sharp}\pi_1(X_K)\)

@@ 18번째 줄: / 18번째 줄: @@
 covering <math>p:Y \to X</math>
-Deck transformation group  = <math>\pi_1(X)/p_{\sharp}\pi_1(Y)</math>
+Deck transformation group  =
+<math>\pi_1(X)</math> is detecting the information between <math>X</math> and its universal covering.
 If this is regular covering, then the group on the right side is same as the Galois group.
@@ 32번째 줄: / 34번째 줄: @@
 this is analogous to the universal covering.
-Suppose we have field extensions
+Suppose we have field extensions <math>\bar{F} , K, F</math>.
+We associate some imaginary space <math>X_F</math> to the field <math>F</math>.
+And get a projection <math>p:X_K \to X_F</math>
+Then we define the absolute Galois group <math>\text{Gal}(\bar{F}/F)=\pi_1({X_F})</math> as a homotopic concept.
+<math>\text{Gal}(K/F)=|K:F|</math>
@@ 211번째 줄: / 223번째 줄: @@
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 * 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
+<math>\text{Gal}(K/F)=\pi_1(X_F)/p_{\sharp}\pi_1(X_K)</math>