By Huishi Li

ISBN-10: 9812389512

ISBN-13: 9789812389510

Designed for a one-semester path in arithmetic, this textbook offers a concise and sensible creation to commutative algebra when it comes to basic (normalized) constitution. It exhibits how the character of commutative algebra has been utilized by either quantity idea and algebraic geometry. Many labored examples and a few 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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It follows from part (ii) that q ( t ) (assuming monic) is the minimal polynomial of z over K ( h ) . Thus, I3 [ K ( z ): K ( h ) ]= degq(t) = max{degu(s), degv(z)}, as desired. 15. Corollary (i) Let E be any intermediate extension field of K with K E C K ( z ) . Then [ K ( z ): El < co. (ii) Every automorphism of the ring K ( z ) which is K-linear is given by Proof Exercise. 16. Theorem (Luroth) Let K be a field and z a transcendental element over K . Let E be an intermediate extension field of K with K $ E C K ( z ) .

Of a over K and the splitting field of p a ( % ) . , rn. Then nzl(z el + e2 + . . + em = n = degp(s), and each a E L = K ( 8 ) is associated to a monic polynomial in E [ z ]that , is, For convenience, we call fa(z)the total polynomial of a. 2. Proposition Let K K ( 6 ) ,the following hold: L = K ( 6 ) c E be as above. For any a E L = ( 9 f a ( z ) E Wzl. ) 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 .

8. Let p be a prime number. Show that xn - p is irreducible in Z[x]and hence in Q[x]. 9. Prove that f = l l y x 8 3y7x5 9x5 - 7y7 - 21 is irreducible in Z[x,y ] . ) + + 3. Field Extensions The study of field extensions stems from the study of zeros of polynomials and the study of irreducibility of polynomials. ,x,] a polynomial of degree 2 2. A full demonstration of this aspect is given in Chapter 4 and Chapter 5. In this section we focus on several fundamental topics concerning field extensions.

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An Introduction to Commutative Algebra: From the Viewpoint of Normalization by Huishi Li


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