103b_06w_2e - 3 A ” and NOT as “ Q a where a is a zero...

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Math 103B Second Hour Exam (50 points) 8 March 2006 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for two pages of notes. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (4 pts.) Find a multiplicative inverse of 1 + 2 x in Z 4 [ x ]. You must do the calculations that show your answer is a multiplicative inverse. 2. (4 pts.) Compute the Hamming distance between the two words u = 0 1 0 0 0 1 0 1 and v = 1 1 1 1 0 0 1 1. Also, either (a) find one word that is simultaneously within Hamming distance two of both u and v or (b) explain why there is no such word. 3. (6 pts.) Find the splitting field of x 3 2 over Q . You should use real and/or complex numbers in your description of the field. For example, give the splitting field of x 2 3 over Q as “ Q ( 3),” NOT as “ Q [ x ] / ( x 2
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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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