One of the recurring themes of modern number theory seems to be a nagging feeling that almost all the things we know (and many more things we don’t know) can be expressed as elementary properties of and -functions.

For example, the zeta function of a number field is defined by the product

If we expand this as a Dirichlet series

the coefficients give detailed information about the factorisation of ideals in . For example, if then is totally ramified, whereas if it is equal to the degree of the field, is totally split.

It is also well-known that the Riemann zeta function (the case above) controls the distribution of the prime numbers in . By modifying this function we can obtain the Dirichlet L-functions which give us even more information. To define these, we start with a homomorphism called a Dirichlet character, and `twist’ the -function using it:

For example, the trivial character recovers the original -function. Dirichlet famously used this, together with observations about the (lack of) zeroes of these (mostly) convergent power series at and , to deduce that any arithmetic progression which might plausibly contain infinitely many primes does. Note that if is the least quotient factors through, it is called the conductor of .

It seems to be true that certainly for an abelian extension, the above two objects are related, and in fact can be written as a product of Dirichlet L-functions. I don’t know enough class field theory to go into the general cases, but want to look at a specific case that seems quite interesting. Apparently the following result is true. Let be a fixed prime, and any quadratic (taking values and ) character with conductor . Now consider the quadratic field . We have the following identity:

Looks innocuous enough…. surely even if this is true, it can’t say much interesting.

Well, let’s unravel what it says:

So cancelling the familiar factors from , it says that

So comparing product expansions, it is equivalent to

Interesting. This splitting into cases reminds us of a more classical area of algebraic number theory. Write , and assume . Then the minimal polynomial is , which factorises modulo an odd prime iff it does after we’ve multiplied by 4 and completed the square to get , whose factorisation modulo , together with the Kummer-Dedekind theorem for the splitting of primes in number fields whose ring of integers is primitively generated, gives a similar criterion in terms of the Legendre symbol:

Now, take and, noting that , we see straight away, comparing the above two statements, that:

So our identity in terms of and L-functions was actually a (fairly heavily) disguised statement of Gauss’ famous law of quadratic reciprocity! Of course, I haven’t given you a proof: I’m not personally sure exactly how to prove the identity about the and L-functions. Maybe I’ll let you know when I figure it out, but in the meantime there are plenty of proofs of quadratic reciprocity out there: http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html. I should also refer the interested reader to Kolowski’s opening chapter of `an introduction to the Langlands program’ which was certainly where I read about this.

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