Thread: Finite fields....

Hi everyone,

I'm trouble solving the following question. (note - calculator not allowed)

Prove that this only 1 distinct intermediate subgroup between K = finite field of order 2 adjoined with Theta a root of X^4+X^3+1 & the finite field of order 2.

I attempted to show that the elements fixed by sigma, sigma^2 & sigma^3 are the same (where sigma is the Frobenius map A |---> A^2 ), but I just get lost in calculation! :-s

Is there an easier method?

Many thanks. x

Originally Posted by AAM

Hi everyone,

I'm trouble solving the following question. (note - calculator not allowed)

Prove that this only 1 distinct intermediate subgroup between K = finite field of order 2 adjoined with Theta a root of X^4+X^3+1 & the finite field of order 2.

I attempted to show that the elements fixed by sigma, sigma^2 & sigma^3 are the same (where sigma is the Frobenius map A |---> A^2 ), but I just get lost in calculation! :-s

Is there an easier method?

Many thanks. x

The polynomial is irreducible over . Let be a root in some larger extension field. Construct . Now and where is the Frobenius automorphism i.e. by . Therefore, the Galois group is cyclic. The intermediate subfields of degree 2 over correspond to subgroup of the Galois group of index 2. However, and so there is only one subgroup of index 2.