Algebraic Theory of Quadratic Numbers
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By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes.
The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.
multiplication: if a ∈ R and x ∈ I, then ax ∈ I. Most of the ideals in this book will be of the form for some , termed generators. 2.2.2 Example. We will show that absorbs multiplication by . It suffices to check that , which we check on the generators: We’re also interested in functions that respect the ring structure. 2.2.3 Definition. Given rings R and S, a function φ: R → S is a ring homomorphism if it meets the following conditions for any a,b ∈ R: (a) (b) φ(ab) = φ(a)φ(b) (c)
f(q) = q; and (b) . If you’re familiar with Galois theory you will recognize this proof as a special case. 4.2 The Ring of Integers We restate Def. 1.3.3 in the context of a general quadratic field. 4.2.1 Definition. The ring of integers of is the set The second equality is Prop. 4.1.3 (c). Though defined as a set, it is clear from the following explicit description that is indeed a ring. 4.2.2 Theorem. Assume that is square-free. The set of integers in is a ring given by , where
too big. But has no solution in , which disqualifies 16 as a candidate for . Next, let’s try to solve . As before, we can’t have , which leads us to a solution (x, y) = (4, 1), and the factorization . Thus, . The moral of the preceding example is that, for imaginary F, solving the equation is easy. Since D F < 0, the norm increases with x and y, and we only need to check finitely many values of either. 5.4.12 Example. For real quadratic fields the situation is harder, as we will see in the
∗ Compute for . Use Cor. 6.6.4 to decide when ideals are principal, as in Ex. 6.6.5. 6.6.5. Conclude Exer. 5.4.6 (c): show that . 6.7 The Group of Units of a Real Quadratic Field So far, we’ve looked at continued fractions as a tool for approximating real numbers. But they have another, unexpected use: determining the group of units in the ring of integers of a real quadratic field F. In Exer. 4.2.4 we saw that there is some leeway in choosing δ. In the context of real quadratic fields,
For the converse, assume that q(η, 1) = 0 and . By the first condition, η = η q or . The latter is impossible, since . We denote the inverse of by . This allows us to concisely extend the definition of a discriminant from quadratic integers (Def. 4.2.3) to an arbitrary quadratic numbers. 7.3.4.Definition For , we put . If with relatively prime, it’s easy to check that 7.3.5.Example A direct computation shows that In general, we have . The discriminant of is47. For a general way of