By Kenneth Ireland, Michael Rosen

This well-developed, available textual content information the old improvement of the topic all through. It additionally presents wide-ranging insurance of important effects with relatively straightforward proofs, a few of them new. This moment variation comprises new chapters that offer a whole evidence of the Mordel-Weil theorem for elliptic curves over the rational numbers and an summary of contemporary growth at the mathematics of elliptic curves.

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**Extra resources for A Classical Introduction to Modern Number Theory**

**Example text**

The theorem of Dirichlet mentioned above has been proved for k[x], k a finite field, by H. Kornblum [50]. Kornblum had his promising career cut short after he enlisted as Kriegsfreiwilliger in 1914. The prime number theorem also has an analog in k[x]. This was proved by E. Artin in his doctoral thesis [2]. A good introduction to analytic number theory is Chandrasekharan [112]. In the last chapter of this very readable text a proof of the prime number theorem is given that uses complex analysis.

Put n(x) = m so that Pm+ 1 > x. If S = {Pl' ... , Pm} then clearly fs(x) = X which implies that X ::;; 2m Jx = 2*) Jx. The result folIows immediately. O It is interesting to note that the above method can also be used to give another proof to Theorem 2. For if L 11Pn converged then there is an n such that Lj>n 11pj < t. If S = {Pl' ... , Pn} then X - fix) is the number of positive integers m ::;; x with y(m) Cs;: S. That is, there exists a prime Pj,j > n such that Pj Im. For such a prime there are [xlpj] multiples of Pj not exceeding x.

There are exactly cjJ(m) units in Z/mZ. Z/mZ is afield iffm is a prime. Corollary 1 (Euler's Theorem). lf(a, m) = 1, then a