Unformatted text preview: 3 A ” and NOT as “ Q ( a ) where a is a zero of x 2 − 3.” 4. (18 pts.) Let E = Q ( √ 2 + √ 5) and F = Q ( √ 10). (a) Prove that F is a sub±eld of E . (b) Find a basis for E as a vector space over F . You need not prove that it is a basis. (c) Find a basis for E as a vector space over Q . You need not prove that it is a basis. 5. (18 pts.) Suppose F and K are ±elds and that F is a ±nite ±eld of characteristic p . (a) Describe explicitly all the values that  F  can have. For example, DO NOT say “the size of any ±eld with characteristic p . If it were correct (which it is NOT), you could say something like “ p and p 2 − 1.” (b) Prove: If K is a ±nite extension of F , then  K  =  F  n for some integer n . (b) Prove: If  K  =  F  n for some integer n , then K is a ±nite extension of F . END OF EXAM...
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 Spring '08
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 Math, Algebra, Determinant, pts, Complex number, Multiplicative inverse, Hamming distance

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