John of Evolving Thoughts asks ScienceBloggers to explain what their scientific fields are all about. I’m not a ScienceBlogger, but given that the only math person on ScienceBlogs is the comp sci-oriented Mark Chu-Carroll, I think it’s a good idea for me to explain my own field.
Number theory is the study of integers, primes, and integer equations. A classical number theoretic question is, what can we say about the number of primes less than x? The classical result is that that number, p(x), is approximately x/ln x, where ln is just a log taken to base e = 2.71828…
The distribution of primes is important mostly in analytic number theory, which uses methods from calculus and analysis to answer questions (the original proof that p(x) ~ x/ln x is based on a function on complex numbers). In algebraic number theory, the original motivating questions were Fermat’s last theorem, and the higher reciprocity laws.
First, Fermat’s last theorem. Algebraic number theory’s basic idea is to use algebraic numbers – that is, numbers that are solutions to some polynomial equation with integer coefficients, like SQRT(2) – to attack problems. To prove that there are no integer solutions to x^n + y^n = z^n for n > 2, mathematicians factored x^n + y^n using algebraic numbers.
For a relatively simple example, take n = 3. In ordinary integers, we can only factor x^3 + y^3 as (x + y)(x^2 – xy + y^2). But suppose that we have another number, w, which has the property that w^3 = 1 but w != 1. We can then factor x^3 + y^3 as (x + y)(x + wy)(x + (w^2)y); the fact that all factors are linear is of tremendous value.
And second, reciprocity laws. We say that p is a quadratic residue of q if p is congruent to a square modulo q. Clearly, if p is square then it’s a quadratic residue, but it doesn’t have to be a square. 2 is a quadratic residue of 7, since 2 = 3^2 mod 7; however, 2 is not a square.
At the end of the 18th century, Gauss proved the law of quadratic reciprocity: if p and q are odd primes congruent to 3 mod 4, then exactly one is a quadratic residue of the other; if p and q are odd primes of which at least one is congruent to 1 mod 4, then either each is a quadratic residue of the other, or each is a quadratic non-residue of the other.
A natural question is then, is there any similar theorem for cubic, biquadratic, quintic, or even higher reciprocity? To answer the question, it’s necessary to work with algebraic numbers.
In both cases, Kummer encountered the same stumbling block in the middle of the 19th century: unique factorization. In ordinary integers, we can always factor numbers into primes uniquely. In some algebraic extensions, we still can, while in others we can’t. For example, if we work in integers of the form a + b*SQRT(-5), where a and b are ordinary integers, then 6 can be factored as 2*3 or (1 + SQRT(-5))(1 – SQRT(-5)); none of the four factors has any divisor except itself and 1, so it’s not like factoring 12 as 2*6 or as 3*4.
Although neither of the two motivating problems is especially important to number theorists, the impetus they’ve provided is immense. Wiles proved Fermat’s last theorem by proving a far more important theorem, that of Taniyama and Shimura. A sweeping generalization of the higher reciprocity laws has led to the Langlands conjectures, of which the Taniyama-Shimura theorem is but a special case.