calculating the splitting of primes
Let be an extension of number fields, with rings of integers . Since this extension is separable (http://planetmath.org/SeparablePolynomial), there exists with and by multiplying by a suitable integer, we may assume that (we do not require that . There is not, in general, an with this property). Let be the minimal polynomial of .
Now, let be a prime ideal of that does not divide , and let be the reduction of mod , and let be its factorization into irreducible polynomials. If there are repeated factors, then splits in as the product
For example, let where is a square-free integer. Then . For any prime , is irreducible mod if and only if it has no roots mod , i.e. is a quadratic non-residue mod . Using quadratic reciprocity, we can obtain a congruence condition mod for which primes split and which do not. In general, this is possible for all fields with abelian Galois groups, using field .
Furthermore, let be the splitting field of . Then acts on the roots of , giving a map , where . Given a prime of , the Artin symbol for any lying over is determined up to conjugacy by . Its in is a product of disjoint cycles of length where . This is useful not just for prime splitting, but also for the calculation of Galois groups.
Another useful fact is the Frobenius theorem, which that every element of is for infinitely many primes of .
For example, let . This is irreducible mod 3, and thus irreducible. Galois theory tells us that is a subgroup of , and so is isomorphic to or , but it is not obvious which. But if we consider , , and the quadratic factor is irreducible mod 7. Thus, .
Or let for some integers and is irreducible. For a prime , consider the factorization of . Either it remains irreducible ( contains a 4-cycle), splits as the product of irreducible quadratics ( contains a cycle of the form ) or has a root. If is a root of , then so is , and so assuming , there are at least two roots, and so a 3-cycle is impossible. Thus or .
|Title||calculating the splitting of primes|
|Date of creation||2013-03-22 13:53:24|
|Last modified on||2013-03-22 13:53:24|
|Last modified by||mathcam (2727)|