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Number Theory

Wonders of numbers - download pdf or read online

By Pickover C.A.

ISBN-10: 0195133420

ISBN-13: 9780195133424

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This approach has been transplanted to number theory through analogies and proved to be effective. Moreover, the Iwasawa conjecture, which we will discuss in Num­ ber Theory 5, came into existence on the basis of a curious analogy between algebraic number fields and algebraic function fields of one variable. Comparing these two kinds of fields, it is generally easier to deal with algebraic function fields of one variable. In order to study a problem in an algebraic number field, it is quite common and often effective to study a corresponding problem in algebraic function fields of one variable as a model case.

3) Given an integral domain A, the following three conditions are equivalent. (i) A is the valuation ring of a discrete valuation of the field of fractions of A. (ii) A is a principal ideal domain with only one nonzero prime ideal. (iii) A is a Dedekind domain with only one nonzero prime ideal. P roof . (1) Let A be the valuation ring of a discrete valuation V, a an element of K such that v{a) = 1. Take a nonzero ideal a of A and set n = min{z/(x) |x G a}. Then we easily see a = (a^) = {x G K \v(x) > n}.

30, we see that p is totally decomposed in AT(a). 8 that p = 1 mod 8. 32. Let K = (V -2 6 ) and a = 0 ^ . In this case it is known that K{a) is not an abelian extension of Q. 27, we have for primes p ^ 2,13, There exist x, y G Z such that p = x^ + 26y^ p is totally decomposed in ii(a ). 10, we cannot rephrase this statement in the form p = •••mod N no matter which natural number N we may take. 1. Questions such as whether or not a prime number p can be written in the form x^ + 6y^, or whether or not p is a prime factor of a number of the form x^ -h 6, are related to the way p decomposes in an algebraic number field.

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Wonders of numbers by Pickover C.A.


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