Algebraic Number Theory (Springer Undergraduate Mathematics Series)
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The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic.
Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform.
The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.
embedding extends to an embedding in ways; by definition of an extension of embeddings, each extension maps to . We can do this for each of the embeddings , extending each in ways. We thus obtain embeddings from to . But we know that there should be exactly embeddings from into , again by Proposition 3.7. We therefore conclude that all of the embeddings have been obtained, and we have seen that is taken to each of its conjugates with multiplicity . Corollary 3.12 Suppose in has minimal
follows that the characteristic polynomial of on is given by . But the roots of , by definition, are exactly the conjugates of . The roots of are therefore the conjugates of , taken with multiplicity . By Proposition 3.11, these are exactly the images of under all the embeddings , and the result then follows. Corollary 3.17 If , then and are both in . Proof As , its minimal polynomial . With the notation of Corollary 3.12, we see that . But this implies that the product ; the constant
value in . As , certainly then contains all multiples of ; that is, . Conversely, take . We can write by Definition 4.28, where either or . As and , we conclude that . But if , then this contradicts the choice of as an element of with the least possible value of the Euclidean function. So , and therefore ; thus every element of is a multiple of , and is principal. The converse to this is false, but it is not so easy to write down an example. In fact, if , then it is known that is a PID, but
absolute value of the middle coefficient gets smaller. Let’s see an example of this procedure in action. Consider the form , or in our abbreviated notation. This is not reduced, but we do have , as . But we do not have . We therefore apply the first rule, to get (that is, our original form is properly equivalent to ). Now we have a form which is not reduced because . We therefore apply the third rule , and find (so now our original form is properly equivalent to ). Now is not reduced as , so
algebraic number dividing , and so , a contradiction. So if is a nontrivial factor of , and therefore of , and is a primitive th root of unity which is a root of , then all powers must be roots of for all coprime to ; simply factor into primes, and apply the result above successively. In particular, every primitive th root of unity is a root of , showing that , i.e., that is irreducible. Corollary 9.9 If is a primitive th root of unity, then . Exercise 9.2 x is odd, . [Hint: show that is