## Geometric Intuition

Let be a number field, and let be the ring of integers of . To employ our geometric intuition, as the Lenstras did on the cover of [LL93], it is helpful to view as a 1-dimensional scheme all prime ideals of $O_K$ over is prime There is a natural map that sends a prime ideal to . For example, if then . For more on this viewpoint, see [Har77] and [EH00, Ch. 2].

If is a prime number, then the ideal of factors uniquely as a product , where the are maximal ideals of . We may imagine the decomposition of into prime ideals geometrically as the fiber , where the exponents are the multiplicities of the fibers. Notice that the elements of are the prime ideals of that contain , i.e., the primes that divide . This chapter is about how to compute the and .

Remark 4.1.1   More technically, in algebraic geometry one defines the inverse image of the point to be the spectrum of the tensor product ; by a generalization of the Chinese Remainder Theorem, we have William Stein 2012-09-24