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splitting of primes

asked 2011-09-27 09:17:08 -0500

Menny gravatar image

updated 2011-09-27 18:59:11 -0500

Mike Hansen gravatar image

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

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answered 2011-09-29 05:25:49 -0500

parzan gravatar image

updated 2011-09-29 05:48:24 -0500

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
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Comments

@parzan - you're the man!

Menny gravatar imageMenny ( 2011-10-02 01:14:15 -0500 )edit

no you!

parzan gravatar imageparzan ( 2011-10-02 04:25:44 -0500 )edit
1

answered 2013-12-19 21:36:55 -0500

John Cremona gravatar image

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).

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Asked: 2011-09-27 09:17:08 -0500

Seen: 595 times

Last updated: Dec 19 '13