I’m not going to prove that every ring of integers of a number field is a Dedekind domain here, but I promise to do so in the next post.
Recall that if K is a number field of degree n, then there are exactly n distinct ring homomorphisms from K to C. I’ll denote them by w1, w2, …, w(n), where w1 is the natural inclusion. For instance, if K = Q(2^(1/3)), then w1 will send 2^(1/3) to itself, and w2 and w3 will send 2^(1/3) to t*2^(1/3) and (t^2)*2^(1/3) where t^3 = 1 but t != 1.
Those of you who have had the misfortune of taking linear algebra in college will know that there exists something called a matrix and a determinant of a matrix. Let A be any n*n matrix, with entries in a field, say C. It’s convenient to look at the rows or columns of A as vectors in C^n. Also, let a(ij) or a(i, j) be the element in row i and column j.
The determinant of A is one function of the entries of A. If n = 1, then |A| is just a11, the only entry of A. If n > 1, |A| is gotten by expanding along some row or column of A (it doesn’t matter which row or column). For example, taking the first row, it’s equal to a11*B11 + a12*B12 + … + a(1n)*B(1n). B(ij) is equal to the determinant of the (n-1)*(n-1) matrix we get if we remove the ith row and jth column from A, and multiply by -1 if i+j is odd.
More concretely, if A is 2*2, we can write it as [a11, a12; a21, a22]. Expanding along the first row, we get |A| = a11*B11 + a12*B21 = a11*|a22| + a12*(-1)*|a21| = a11*a22 – a12*a21.
The determinant has the following properties, none of which I’m going to prove. If A’ is gotten from A by switching any two rows, or switching any two columns, then |A’| = -|A|. If A’ is gotten from A by adding a multiple of one row to another row, or a multiple of one column to another column, then |A’| = |A|. If A’ is gotten from A by multiplying a single column or a single row by a constant k, then |A’| = k|A|. |A| = 0 iff the row vectors of A are linearly dependent over a field containing all a(ij)’s, which is true iff the column vectors of A are linearly dependent over such a field.
Going back to algebraic number theory, given any list of n elements of K, (x1, x2, …, x(n)), we define their discriminant to be |A|^2, where the element a(ij) of A is given by w(i)(x(j)). |A|^2 is clearly invariant under switching rows or switching columns, so it doesn’t matter how we order the x(j)’s or the w(i)’s. Also, it’s sometimes useful to talk about disc(x) = disc(1, x, x^2, …, x^(n-1)).
The discriminant satisfies the following properties:
- It is always rational. It’s clearly contained in K’, the smallest number field containing all conjugates of K. Every ring homomorphism on K’, say F, will permute the w(i)’s, so it will act on A by switching around rows. Since |A|^2 is invariant under switching rows, we get |A|^2 = |F(A)|^2 = F(|A|^2) for all F. In a way it turns |A|^2 into a symmetric polynomial in the conjugates of a (where K = Q(a)), so it will be rational.
- If the x(j)’s are algebraic integers, then |A|^2 is an integer, because it’s the sum of products of algebraic integers.
- |A|^2 can be negative; that is, |A| can be purely imaginary. If K = Q(i), then disc(i) = |A|^2, A = [1, i; 1, –i], so |A| = -2i and |A|^2 = -4.
- |A| = 0 if the x(j)’s are linearly dependent over Q. If a1x1 + a2x2 + … + a(n)x(n) = 0, then for every ring homomorphism w, w(1) = 1, so by addition and division, w(p/q) = p/q for integers p and q; so 0 = w(a1x1 + a2x2 + … + a(n)x(n)) = a1w(x1) + a2w(x2) + … + a(n)w(x(n)). Then letting c(j) be the jth column, we get that a1c1 + a2c2 + … + a(n)c(n) = 0 and the a(n)’s are not all 0.
- |A| = 0 only if the x(j)’s are linearly dependent over Q. If a1c1 + a2c2 + … + a(n)c(n) = 0, we want to show that the a(j)’s are in Q. But they’re invariant under every ring homomorphism of K’, which will just switch around rows, so they must be in Q.
- From 5, if K = Q(a), then disc(a) is not zero.