By Huishi Li

Designed for a one-semester path in arithmetic, this textbook provides a concise and functional creation to commutative algebra by way of general (normalized) constitution. It exhibits how the character of commutative algebra has been utilized by either quantity thought 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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3. Definition For Q E L , TL/K(Q)is called the trace of NL,K(Q) is called the norm of CY in K . 4. Proposition For Q,P E L , X,p E K , the following hold: + (i) TL/K(XCY + PP) = XTL/K(CY)/-LTL/K(P). 2 is given as an appendix a t the end of this section. 5. , 8,, then the bilinear form defined above is nondegenerate. , n - 1, and write V for the Vandermonde matrix [ ;;;; r(0) V= r ( n - 1) 1 1 ... 1 :; :; ::: :: 6:-' q - 1 ... . ,n. , c cs p t i= 1 i=l n T,/,(6k@) = Ci(6k6j)= n = d(Ic)(6(j))t, the matrix of the bilinear form is given by ( T L / K ( d W ) ) = VVt and hence det ( T ~ / ~ ( s ~= d j(det(V))2.

L , . Exercises 1. Show that Z[i] = { a bi [ a, b E Z}Z Z@ Z. 2. 4. 3. 5. 4. An abelian group G is said to be torsion-free if G does not have finite order nonzero element. Show that a finitely generated torsion-free abelian group is free of finite rank. ) 5. Show that a finitely generated abelian group G is either finite or isomorphic to the direct sum of a free abelian group of finite rank and a finite abelian group. + 7. Noetherian Modules Let R be a ring. Definition Let M be an abelian group with the binary additive operation + and the identity element 0.

Let M be an R-module and suppose that IM = @iEJMi for some submodules Mi c M , i E J . , C mi = 0 if and only if mi = 0. 2. Definition An R-module M is said to be free if there are E M , i E J , such that M = @ i E J R J ~ ,where {*
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