# Revision history [back]

The following works for me:

S.<X> = QQ[]
R.<x> = K[]

polynomials = [X^2 + 1, X^2 + 2, X^2 + X + 1]

count = 0
for f in polynomials:
count += 1
print "[%s]" % count
if R(f).is_irreducible():
print "\tIrreductible case for %s over K..." % f
L = NumberField( [X^2+1, f], names='a');
print "\tThe number field L is:"
print "\t%s" % L
print "\tThe base field of L is:"
print "\t%s" % L.base_field()
print "\tL has absolute degree %s" % L.absolute_degree()
print "\tL has relative degree %s" % L.relative_degree()
else:
print "\tReducible case for %s over K" % f


Results:


Reducible case for X^2 + 1 over K

Irreductible case for X^2 + 2 over K...
The number field L is:
Number Field in a0 with defining polynomial X^2 + 1 over its base field
The base field of L is:
Number Field in a1 with defining polynomial X^2 + 2
L has absolute degree 4
L has relative degree 2

Irreductible case for X^2 + X + 1 over K...
The number field L is:
Number Field in a0 with defining polynomial X^2 + 1 over its base field
The base field of L is:
Number Field in a1 with defining polynomial X^2 + X + 1
L has absolute degree 4
L has relative degree 2


Note that f.is_irreducible is only a method, as a boolean it always avaluates to True, one has to call it, f.is_irreducible() to have the needed True or False. Also, always make the difference between the transcendentals like x over different fields.