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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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This note was uploaded on 06/18/2009 for the course MATH 103b taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Math, Algebra

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