By Gerald J. Janusz
The ebook is directed towards scholars with a minimum history who are looking to research category box concept for quantity fields. the one prerequisite for studying it really is a few common Galois conception. the 1st 3 chapters lay out the mandatory historical past in quantity fields, such the mathematics of fields, Dedekind domain names, and valuations. the following chapters talk about classification box idea for quantity fields. The concluding bankruptcy serves as an example of the options brought in past chapters. specifically, a few attention-grabbing calculations with quadratic fields exhibit using the norm residue image. For the second one version the writer further a few new fabric, extended many proofs, and corrected blunders present in the 1st version. the most target, notwithstanding, is still similar to it used to be for the 1st version: to provide an exposition of the introductory fabric and the most theorems approximately type fields of algebraic quantity fields that will require as little historical past training as attainable. Janusz's e-book will be a great textbook for a year-long direction in algebraic quantity idea; the 1st 3 chapters will be compatible for a one-semester direction. it's also very appropriate for self reliant examine.
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Extra resources for Algebraic Number Fields
Let R' denote the integral closure of the Dedekind ring R in a finite-dimensional extension L of the quotient field K of R . Let p be a nonzero prime ideal of R . Suppose there is an element 6 such that the integral closure of R, in L is R,[B]. LetJ'(X) be the minimal polynomial of 0 over K . Letf(X) denote the polynomial obtained by reducing the coefficients off'(X) modulo p. Suppose f ( X ) = g,( X p. * . g , ( X Y is the factorization off(X) as a product of the distinct irreducible polynomials 33 7.
X,) = det IT(xixj)I is called the discriminant of the basis xI..... x,. If we select the xiin R', then x i x j is in R ' s o T ( x i x j )isin R . ,x, range over all possible bases of L / K which lie in R', the discriminants generate an ideal of R which we shall call the discriminant ideal of R' over R . We denote this ideal by A or A ( R ' / R ) . We begin the study of the discriminant by showing it can be determined by localization. 1 Lemma. Let S be a multiplicative set in R . Then A(Rs'/Rs) = A ( R ' / R ) s .
Let p i = q in R . I) = pyif'. Consider now the case with K = Q = rational field and R = Z = rational integers. L and R' have the same meanings as above. For any ideal 91 of R', N (91) is an ideal in Z which is necessarily a principal ideal, say N (91) = Zm = (m)for some integer m. If we require that m 2 0 then m is uniquely determined. Let us denote the integer m by N(91)so that the norm of an ideal 91 # 0 is now a positive integer. I. 6 Proposition. I) is equal to the number of elements in the ring R'/21.