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Find a subfield of order 4 in E.

Consider the irreducible polynomial g=X^4+X+1 over F2 and let E be the extension of F2={0, 1} with a root α of g.

Find a subfield of order 4 in E.

Im really not sure about this question as ive found the elements in E being {0,1,α,α^2,α^3,1+α,1+α^2,1+α^3,α+α^2,α+α^3,α^2+α^3,1+α+α^2,1+α^2+α^3,1+α+α^3,α+α^2+α^3,1+α+α^2+α^3} but cant seem to answer the original question.

Any help will be appreciated
(edited 7 years ago)
Original post by AFraggers
Consider the irreducible polynomial g=X^4+X+1 over F2 and let E be the extension of F2={0, 1} with a root α of g.

Find a subfield of order 4 in E.Find a generator h for the non-zero elements under multiplication (the obvious guess works, I think). Then H = h^5 iis a cube root of unity, as is H^2. Then I think {0, 1, H, H^2} should be your subfield.

You'll need to verify it's closed under addition, but in any subfield of order 4 the there are 3 non-zero elements and so they form a group of order 3, so the subfield must contain the 3 cube roots of unity.

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