By Huishi Li
Designed for a one-semester direction in arithmetic, this textbook provides a concise and useful creation to commutative algebra when it comes to common (normalized) constitution. It indicates how the character of commutative algebra has been utilized by either quantity thought and algebraic geometry. Many labored examples and a couple of challenge (with tricks) are available within the quantity. it's also a handy reference for researchers who use simple commutative algebra.
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Extra resources for An Introduction to Commutative Algebra: From the Viewpoint of Normalization
Thus, f vanishes when 21 = 5 2 . ,xn][xl] yields (21 -xz)lf. , n. ,x,] and is antisymmetric. ) 5 . Trace and Norm s Throughout this section we let K L be a simple algebraic field extension, that is, L = K ( 6 ) , 6 E L. ,6,, where m 5 n = [ L : K] = degp(x), that is, E contains the splitting field of p(x). 1. Proposition With notation as above, there are exactly m distinct ring monomorphisms L -+ E that are K-linear. Moreover each K-linear monomorphism L --f E is given by 6 4 6i, 1 5 i 5 m. , ~ ( 1 9 )is a zero of p(x).
G S } forms a basis for H , as desired. 7. Theorem Let G be a free abelian group of rank n and H a subgroup of G. The following statements hold: 44 Commutative Algebra (i) G I H is finite if and only if rankG = rankH. , y n } is a then the number of elements of G I H is equal t o Idet(A)I, where A = ( a i j ) . , s 5 n. Thus and we have the group isomorphism It follows that G I H is finite if and only if n = s. en elements. /! . 0 0 ... en Taking the matrix A = ( a i j ) from the assumption of part (ii) into account, we get A = BCD and det(A) = det(B)det(C)det(D).
E K [ z ]be the minimal polynomial of a over K . Then fa(z)= pa(z)' for some s 2 1. ) of a over K . Proof (i) Since a = ~ ( 2 9 ) = Cyi: A@, where we have i=1 T(X) = CyL; Xixi E K [ z ] , Preliminaries 35 Note that all X i E K . After expanding the latter product we see that the coefficients of fa(z)are given by symmetric polynomials in the n zeros of p ( z ) . 2, fa(z)E K [ z ] . ) E K [ z ]and f a ( a )= 0. It follows that fa(z)= p a ( s ) s h ( z )where , h ( z ) E K [ z ]and p a ( z ) , h ( z ) are coprime and both are monic.