Differential Galois theory

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?

 

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

 

Also the permanence of algebraic relation if 

 

regular covering

covering \(p:Y \to X\)

Deck transformation group  =  \(\pi_1(X)/p_{\sharp}\pi_1(Y)\)

\(\pi_1(X)\) 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.

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

 

 

homotopy lifting theorem

We have a covering map  \(p:Y \to X\).

A loop in X can be lifted into Y.

Then it defines an action on Y.

This action is trivial if this loop in X is the image of a loop in Y.

 

 

 

 

 

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.

 

 

 

 

 

 

하위페이지

• 갈루아 이론
  
  • Abel's impossibility theorem
    
  • adele and idele
    
  • algebraic functions
    
  • Class Field Theory
    
  • Differential Galois theory
    
  • Liouville theory
    
  • number fields and threefolds
    
  • 고전적인 갈루아 이론
    
  • 대수적정수론
    

 

 

 

 

 

 

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작도문제

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표준적인 도서 및 추천도서

• http://gigapedia.info/1/galois_theory
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

참고할만한 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Homotopy_lifting_property
• http://en.wikipedia.org/wiki/covering_space
• http://en.wikipedia.org/wiki/Field_extension
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
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