Dedekind abc abc abba a00aa aa0 abc abc
![실수 (Dedekind) (a+b)+c = a+(b+c), a+b=b+a, a+0=0+a=a, a+(-a)=0 (ab)c= a(bc), ab=ba, 1 a=a 1=a, 실수 (Dedekind) (a+b)+c = a+(b+c), a+b=b+a, a+0=0+a=a, a+(-a)=0 (ab)c= a(bc), ab=ba, 1 a=a 1=a,](https://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-1.jpg)
실수 (Dedekind) (a+b)+c = a+(b+c), a+b=b+a, a+0=0+a=a, a+(-a)=0 (ab)c= a(bc), ab=ba, 1 a=a 1=a, aa-1=1=a-1 a a(b+c) = ab + ac a, b ∊ P ↦ a+b ∊ P; a, b ∊ P ↦ ab ∊ P, a ∊ F a ≠ 0 ↦ a ∊ P 또는 –a ∊ P Dedekind cut
![√ 2√ 2 무리수, 초월수 ? e Euler (1737), Hermite (1873) Pi Lambert(1761), Lindemann(1882) √ 2√ 2 무리수, 초월수 ? e Euler (1737), Hermite (1873) Pi Lambert(1761), Lindemann(1882)](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-2.jpg)
√ 2√ 2 무리수, 초월수 ? e Euler (1737), Hermite (1873) Pi Lambert(1761), Lindemann(1882) Lambert r 유리수인경우 tan r, er 무리수 Hermite, Weierstrass a가 대수수인경우 ea, log a 는 초월수 Hilbert 7번 문제 a 대수수, b 무리대수수인경 우 ab는 초월수이다. (Gel’fond, Schneider 1934)
![수는 어떻게 주어지는가? Decimal representation Limits of sequences Sums of sequences Infinite product Continued 수는 어떻게 주어지는가? Decimal representation Limits of sequences Sums of sequences Infinite product Continued](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-3.jpg)
수는 어떻게 주어지는가? Decimal representation Limits of sequences Sums of sequences Infinite product Continued fractions Definite integral Etc….
![e= lim n-> (1+1/n)n e = 2. 71828183…. . log x = x 1 e= lim n-> (1+1/n)n e = 2. 71828183…. . log x = x 1](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-4.jpg)
e= lim n-> (1+1/n)n e = 2. 71828183…. . log x = x 1 dt/t, log(e) =1 e = n=1 1/n! e= [2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1…] (Euler)
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![군 group Binary operation *: G x G -> G Associative Identity a * 군 group Binary operation *: G x G -> G Associative Identity a *](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-6.jpg)
군 group Binary operation *: G x G -> G Associative Identity a * e = e * a = a Inverse a * a -1 = e = a-1 * a 예 (Z, +), (Q, +), (R, +), (Q-{0}, *), (R-{0}, *), (C-{0}, *) (Z/n. Z, +), ((Z/n. Z)x, *) 아벨군 a*b = b*a 갈루아
![Dihedral group {1, r, r 2, …, r n-1, s, sr 2, …, sr Dihedral group {1, r, r 2, …, r n-1, s, sr 2, …, sr](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-7.jpg)
Dihedral group {1, r, r 2, …, r n-1, s, sr 2, …, sr n-1}
![Symmetric group A={1, 2, 3, …, n} f: A -> A, g o f: Symmetric group A={1, 2, 3, …, n} f: A -> A, g o f:](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-8.jpg)
Symmetric group A={1, 2, 3, …, n} f: A -> A, g o f: A -> A Cycle decomposition (135)(478)(29) Sn 은 아벨군이 아니다. n>2
![동형군 Homomorphism F: G -> H, F(g*h) =F(g)*F(h) Isomorphism homomorphism + bijection 동형인 것들은 동형군 Homomorphism F: G -> H, F(g*h) =F(g)*F(h) Isomorphism homomorphism + bijection 동형인 것들은](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-9.jpg)
동형군 Homomorphism F: G -> H, F(g*h) =F(g)*F(h) Isomorphism homomorphism + bijection 동형인 것들은 그중 하나만 연구하면 된 다.
![환 Ring (R, +, *) (I) (R, +) abelian group (II) x is associative 환 Ring (R, +, *) (I) (R, +) abelian group (II) x is associative](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-10.jpg)
환 Ring (R, +, *) (I) (R, +) abelian group (II) x is associative (III) distributive law a*(b+c) = a*b + a*c Commutative a*b=b*a Identity
![환의 예 정수 Z, 유리수 Q, 실수 R, 복소수 C Z/n. Z Z( D)={a+b 환의 예 정수 Z, 유리수 Q, 실수 R, 복소수 C Z/n. Z Z( D)={a+b](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-11.jpg)
환의 예 정수 Z, 유리수 Q, 실수 R, 복소수 C Z/n. Z Z( D)={a+b D: a, b Z} Hamiltonian a+ b i + c j + d k i 2=j 2=k 2=-1, ij = -ji=k, jk=-kj=I, ki=-ik=j
![환의 예 다항식환 polynomial ring an x n + b n-1 x n-1 + 환의 예 다항식환 polynomial ring an x n + b n-1 x n-1 +](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-12.jpg)
환의 예 다항식환 polynomial ring an x n + b n-1 x n-1 + … + a 1 x + a 0 행렬환 (aij) (a ij) + (bij) = (aij+ bij) (a ij) (bij) = ( k aik bkj)
![모듈 Module (M, +, R) (M, +) abelian group, R ring R x M 모듈 Module (M, +, R) (M, +) abelian group, R ring R x M](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-13.jpg)
모듈 Module (M, +, R) (M, +) abelian group, R ring R x M -> M (a) (r+s)m= rm + sm (b) (rs)m = r(sm) (c) r(m+n) = rm + rn 1 m = m
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![체(Field) Commutative ring F identity e for *, inverse for F-{0} R, Q Z/p. 체(Field) Commutative ring F identity e for *, inverse for F-{0} R, Q Z/p.](http://slidetodoc.com/presentation_image/2ef74ea828ebb0d0d3bb293d9ee86f06/image-15.jpg)
체(Field) Commutative ring F identity e for *, inverse for F-{0} R, Q Z/p. Z p는 소수 R(x), Q(x), Z/p. Z(x) Field extension K F C = R + Ri R
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