all prime ideals of $O_K$

over

is prime

There is a natural map
that sends a prime ideal
to
.
For example, if

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 .

William Stein 2012-09-24