Given a quadratic field, say,
K.<a>=QuadraticField(2)
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
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
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: K.<a> = QuadraticField(5)
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).
Asked: 2011-09-27 16:17:08 +0100
Seen: 2,027 times
Last updated: Dec 20 '13
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