[현대대수학]노트정리

본문내용

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7. Monic Polynomial (모닉 다항식)
: a field and such that
: a monic polynomial in
8. and
: an extension field of a field
: algebraic over and
: the irreducible polynomial of over
1. : a zero of . i.e.
2.
i.e. has a minimal degree
3. : monic polynomial in
4. : irreducible over
Notation :
9. The Smallest Subdomain, The Smallest Subfield
(최소 부분정역, 최소 부분체)
: an extension field of a field ,
① : the smallest subdomain of
containing and
i.e.
where : subdomain of
If ,
If ,
② : the smallest subfield of
containing and
i.e.
where : subfield of
If ,
If ,
10. Simple Extension (단순확대체)
: an extension field of a field
: a simple extension of
① : algebraic over
where
② : transcendental over
11. Algebraic Extension (대수적 확대체)
: an extension field of a field
: an algebraic extension of
is algebraic over
12. Finite Extension (유한 확대체)
: an extension field of a field
: a finite extension of
: finite dimensional vector space over
※ : an extension field of a field ,
: algebraic over
: finite algebraic simple extension of
13. Algebraic Closure (대수적 폐포)
: an extension field of a field
; the algebraic closure of in
; the algebraic closure of
; a maximal algebraic extension of
14. Algebraically Closed Field (대수적 폐체)
: an algebraically closed field
15. : The Galois Field of order
where ; irreducible over
and
16. Conjugate Element (공액 원소)
: a field,
: conjugate over
17. Conjugation Isomorphism (공액 동형사상)
: a field, : conjugate over
: the conjugation isomorphism for
; a map defined by
18. Fixed Field (고정체)
: a field,
; the fixed field of in
If , ; the fixed field of in
19. Group of over
; 의 원소를 고정시키는 의 자기동형군
( )
20. Index (차수)
: a finite extension of
21. Splitting Field (분해체)
① : a field and and
: the splitting field of over
; the smallest subfield of
containing and
※ 의 모든 다항식의 모든 근과
를 포함하는 가장 작은 체가 분해체이다.
② If
: the splitting field of over
③ : a field and
: the splitting field over
: the splitting field of over
for some
22. Separable Extension (분리확대체)
① : a finite extension of
: a separable extension of
② : separable over
: a separable extension of
23. Perfect Field (완전체)
: a perfect field
: a finite extension of
: a separable extension of
24. Totally Inseparable Extension (완전 비분리확대체)
: a finite extension of
: a totally inseparable extension of
25. Normal Extension (정규 확대체)
: a field and
: a normal extension of
: a separable spitting field over
26. Galois Group (갈로아 군)
: a finite normal extension of
: the Galois group of over
27. Galois Group of Polynomial (다항식의 갈로아 군)
: a field and
: the Galois group of over
: a spitting field of over
28. Cyclotomic Extension (원분 확대체)
: a field and
: the nth cyclotomic extension of
: a spitting field of over
29. Extension by Radical (제곱근에 의한 확대체)
: a field and
: an extension of by radical
for and
such that and
for
30. Solvable by Radical (제곱근에 의한 풀이 가능)
: a field and
: solvable by radical over
where : the splitting field of over
: an extension of by radical

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