Just wanted to draw attention to this very nice exchange on Math Overflow. Matt Emerton remarks that the different of a number field is always a square in the ideal class group, and asks: is there a canonical square root of the ideal class of the different?
What grabs me about this question is that the word “canonical” is a very hard one to define precisely. Joe Harris used to give a lecture called, “The only canonical divisor is the canonical divisor.” The difficulty around the word “canonical” is what gives the title its piquancy.
Usually we tell students that something is “canonical” if it is “defined without making any arbitrary choices.” But this seems to place a lot of weight on the non-mathematical word “arbitrary.”
Here’s one way to go: you can say a construction is canonical if it is invariant under automorphisms. For instance, the abelianization of a group is a canonical construction; if f: G_1 -> G_2 is an isomorphism, then f induces an isomorphism between the abelianizations.
It is in this sense that MathOverflow user “Frictionless Jellyfish” gives a nice proof that there is no canonical square root of the different; the slick cnidarian exhibits a Galois extension K/Q, with Galois group G = Z/4Z, such that the ideal class of the different of K has a square root (as it must) but none of its square roots are fixed by the action of G (as they would have to be, in order to be called “canonical.”) The different itself is canonical and as such is fixed by G.
But this doesn’t seem to me to capture the whole sense of the word. After all, in many contexts there are no automorphisms! (E.G. in the Joe Harris lecture, “canonical” means something a bit different.)
Here’s a sample question that bothers me. Ever since Gauss we’ve known that there’s a bijection between the set of proper isomorphism classes of primitive positive definite binary quadratic forms of discriminant d and the ideal class group of a quadratic imaginary field.
Do you think this bijection is “canonical” or not? Why?