I'm trying to take the degree two tower extension of the complex extension of a field.
I'm new to Sage but I expected F4 to be a field rather than a ring. My question is how to I get the field F4 and not the ring?
P = 2^31 - 1
F = GF(P)
K.<x> = PolynomialRing(F)
F2.<i> = F.extension(x^2+1)
K2.<y> = PolynomialRing(F2)
F4.<u> = F2.extension(y^2-2*i-1)
print(F4)
print((y^2-2*i-1).is_irreducible())
# print(F4.gens()[0].multiplicative_order())
output:
Univariate Quotient Polynomial Ring in u over Finite Field in i of size 2147483647^2 with modulus u^2 + 2147483645*i + 2147483646
True
