Unformatted text preview: F ⊆ E . (b) Probably the simplest basis to ²nd is 1, √ 2 + √ 5; however, there are others such as 1, √ 2. (c) One possibility is 1, √ 2, √ 5, √ 10. 5. (a)  F  must be a power o± p and all powers p k with k a positive integer are possible. (b) I± [ K : F ] = n , then K is a vector space over F o± dimension n and so  K  =  F  n . (c) Suppose  F  = p k and  K  =  F  n = p kn . By the uniqueness o± ²nite ²elds (Theorem 22.1), we have F = GF( p k ) and K = GF( p kn ). By the sub²eld theorem (Theorem 22.3), F is a sub²eld o± K ....
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 Spring '08
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 Math, Linear Algebra, Algebra, Q. • Adjoin

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