# splitting of primes

Is there a method to know if a given prime splits/ stay inert/ ramify in K?

(For number-theoretic background: http://en.wikipedia.org/wiki/Splittin... ) I tried to look it up but didn't find anything.

Thanks a lot!!! Menny

edit retag close merge delete

Sort by ยป oldest newest most voted

Are the ramified primes the ones which divide the field discriminant? If this is so you can get them by

sage: K.<y> = NumberField(x^4 - x^2 + 1)
sage: [x[0] for x in list(K.discriminant().factor())]
[2, 3]

If splitting means that the prime factors then you can check this like this:

sage: is_split = lambda F,x:sum([t[1] for t in list(F.factor(x))])>1

for example:

sage: K.<y> = NumberField(x^2 + 1)
sage: for x in range(30):
if is_prime(x):
print x%4,is_split(K,x)
....:
2 True
3 False
1 True
3 False
3 False
1 True
1 True
3 False
3 False
1 True
more

@parzan - you're the man!

( 2011-10-02 08:14:15 +0200 )edit

no you!

( 2011-10-02 11:25:44 +0200 )edit

For quadratic fields the situation is simpler than the general case. I see no reason why prime ideals should not have (easily implemented) methods is_inert(), is_split(), though in the latter case it is not so clear what is meant for non-Galois extensions. However, prime ideals do know what their residue degree is and their ramification degree (called ramification_index for some reason) so you can do this:

sage: [[(p,P.residue_class_degree(), P.ramification_index()) for P in K.primes_above(p)] for p in prime_range(50)]
[[(2, 2, 1)],
[(3, 2, 1)],
[(5, 1, 2)],
[(7, 2, 1)],
[(11, 1, 1), (11, 1, 1)],
[(13, 2, 1)],
[(17, 2, 1)],
[(19, 1, 1), (19, 1, 1)],
[(23, 2, 1)],
[(29, 1, 1), (29, 1, 1)],
[(31, 1, 1), (31, 1, 1)],
[(37, 2, 1)],
[(41, 1, 1), (41, 1, 1)],
[(43, 2, 1)],
[(47, 2, 1)]]

So ramified prime look like (p,1,2), intert ones (p,2,1) and split ones (p,1,1).

more