"Differential Galois theory"의 두 판 사이의 차이

2009년 7월 31일 (금) 15:21 판

하위페이지

갈루아 이론

재미있는 사실

많이 나오는 질문

네이버 지식인
- http://kin.search.naver.com/search.naver?where=kin_qna&query=갈루아

표준적인 도서 및 추천도서

Abel_s_theorem_by_Arnold.djvu
arnold book on abel theorem problem 348

참고할만한 자료

@@ 1번째 줄: / 1번째 줄: @@
-<h5>간단한 소개</h5>
-There are two important conditions required in the Galois theory.
-* transitivity
-* fixed point free action
-Sometimes, <math>\text{Gal}(K/F)=|K:F|</math> is presented as a condition. This automatically implies fixedpoint free transtive action.
-How important is the transitivity in Galois theory and Monodromy theory?
-<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">dictionary</h5>
-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 theorem in complex analysis corresponds to the theorem of field theory
-If <math>\alpha\in\mathbb{\bar{Q}}</math> is a root of <math>a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 = 0, a_i \in \mathbb{Z}</math>, so is <math>\sigma(\alpha)</math> where the <math>\sigma</math> is the automorphism of the algebraic closure.
-What about the analytic solution of a differential equation and its continuation?
-<h5>How to compute the Galois group</h5>
-* we need to decide where it ramifies, i.e. we need information about the discriminanat.
-* Decide the primes dividing the discriminant.
-* Look how the primes are interchanged.
-*
-<h5>regular covering</h5>
-covering <math>p:Y \to X</math>
-Deck transformation group  =  <math>\pi_1(X)/p_{\sharp}\pi_1(Y)</math>
-<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.
-So we can understand the Galois group from homotopic theoretic viewpoint.
-<h5>the role of algebraic closure </h5>
-this is analogous to the universal covering.
-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.
-Now <math>\text{Gal}(K/F)=\pi_1(X_F)/p_{\sharp}\pi_1(X_K)</math> is defined.
-<h5>homotopy lifting theorem</h5>
-We have a covering map  <math>p:Y \to X</math>.
-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.
-<h5>analogy between the circle and the finite field</h5>
-* the field theoretic analogue of the circle = finite field
-<h5>Galois theory for prime ideals</h5>
-<br>
-<h5>example</h5>
-<math>x^3-2=0</math>
-<math>K=\mathbb{Q}(\omega, \sqrt[3]{2})</math> over <math>\mathbb{Q}</math>
-<math>Spec \mathbb{Z}[\omega,\sqrt[3]2]</math> over <math>Spec \mathbb{Z}</math>
-<math>[K : \mathbb{Q}]=6</math>
-Note that this degree is not equal to the degree of the polynomial.
-<h5>하위주제들</h5>
 ==== 하위페이지 ====
 * [[differential Galois theory|갈루아 이론]]<br>
-** [[책과 자료 파일모음|Abel's impossibility theorem]]<br>
 ** [[3274669|adele and idele]]<br>
 ** [[2086806|algebraic functions]]<br>
@@ 149번째 줄: / 8번째 줄: @@
 ** [[1925124|Liouville theory]]<br>
 ** [[number fields and threefolds]]<br>
-** [[3074078|고전적인 갈루아 이론]]<br>
 ** [[3072420|대수적정수론]]<br>

"Differential Galois theory"의 두 판 사이의 차이

2009년 7월 31일 (금) 15:21 판

목차

하위페이지

재미있는 사실

관련된 단원

많이 나오는 질문

관련된 고교수학 또는 대학수학

관련된 다른 주제들

표준적인 도서 및 추천도서

참고할만한 자료

이미지 검색

동영상

관련기사

블로그

둘러보기 메뉴

검색