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Towered extension fields through chosen polynomial

K. = GF(q)

KT. = K.extension(2)

KTT. = K.extension(6)

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

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

Documentation gives nothing about this towering of fields.

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Towered extension fields through chosen polynomial

K. = GF(q)

GF(q) KT. = K.extension(2)

K.extension(2) KTT. = K.extension(6)

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

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

Documentation gives nothing about this towering of fields.

click to hide/show revision 3
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Towered extension fields through chosen polynomial

K. ...

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

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 ???

Documentation gives nothing about this towering of fields.

click to hide/show revision 4
None

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 ???

Documentation gives nothing about this towering of fields.