현대대수학2 조교

*DEF

algebraic over F

transcendental over F

(monic)irreducible polynomial for α over F

degree of α over F

simple extension of F

vector space over F

span, dimension, linearly independent over F, basis for V over F,

algebraic extension of F

finite extension of F

the algebraic closure of F in E

algebraically closed field

a algebraic closure of F

*THM

Kronecker's Theorem

FEF이면 AEF

AEF가 FEF이다 iff AEF=F(α1,α2,...,ακ) for some α1, α2, ..., ακ in E

About finite fields

Z_p is finite field of order p

About Existence of finite fields

char(F)=p이면 x^p^n - x has p^n distinct zeros in ac(F)

GF(p^n) exists

All finite fields of order p^n are isomorphic

About structures of finite fields

F:finite, E:FEF of F, [E:F]=n

char(F)=p and |F|=p^n for some n and F={x in ac(Z_p) s.t. x^p^n - x in Z_p[x]} and te irreducible poly f(x) in F[x] of degree n

char(E)=p and |E|=|F|^n and E:simple extension of F

따라서 정리하면,

이미 알고 있는 finite fields는 Z_p가 있다.

ac(Z_p)에서 p^n order finite field를 항상 만들 수 있다. GF(p^n)

하지만 임의의 finite field는 더 넓은 집합

임의의 finite field F의 char는 p인걸 알고 따라서 order는 p^n인걸 알고 이것은 GF(p^n)과 isomorphic

기타 부가적인 내용:E:FEF of finite field F일 때 |E|=|F|^n with n=[E:F] and E:simple extension of F

*HW

*Additional Problems

Prove that x^​3 - nx + 2 in Z[x] is irreducible over Z

(n is an integer s.t. it is not equal to -1, 3, 5)

Determine the degree of the extension Q((3+2^​1/2)​^​1/2​) over Q

p:any prime, a:nonzero element in Z_p, Prove that x^p - x + a is irreducible over Z_p

V={0}의 basis는 empty set, 따라서 dimension = 0

x^4 + 1 is irreducible in Z[x], reducible over Z_p for all prime p

char(F)=0이면 E:FEF of F는 simple extension of F

char(F)=p이고 E:FEF of F인데 not simple인 예 존재, F=ac(Z_p)(x^p,y^p), E=ac(Z_p)(x,y), [E:F]=p^2