# Towered extension fields through chosen polynomial

(Edited, since <u> compiles in HTML as an underlyning tag, dan.)

K.<u> = GF(q)
KT.<u> = K.extension(2)
KTT.<u> = K.extension(6)


In this case KTT is a degree 12 extension. My question is:

Can I obtain KTT by KTT.<u> = K.extension(x^6 - a), where x^6 -a is some irreducible polynomial over KT ???

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The following worked for me:

sage: q = 27
sage: K.<u> = GF(q)
sage: K.extension(2)
Finite Field in z6 of size 3^6
sage: KT.<a> = K.extension(2)
sage: KT
Finite Field in a of size 3^6
sage: RT.<Z> = PolynomialRing(KT)
sage: KTT.<b> = KT.extension(Z^6-a)
sage: KTT
Univariate Quotient Polynomial Ring in b over Finite Field in a of size 3^6 with modulus b^6 + 2*a
sage: b.minpoly()
x^6 + 2*a

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