# hmwk5 - be positive integers with gcd m,n = d Show that...

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Math 523 Homework 5 Due 7/28/2008 Total: 20 points From Section 4 of the lecture notes: 1. Give a multiplication table for GF (3 2 ). Find all generators for the cyclic group GF (3 2 ) × , and ﬁnd the minimal polynomial of each generator over Z 3 . 2. Find all generators for the cyclic group of nonzero elements of GF (2 4 ), and ﬁnd the minimal polynomial of each generator over Z 2 . 3. Let m,n
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Unformatted text preview: be positive integers with gcd( m,n ) = d . Show that over any ﬁeld the greatest common divisor of x m-1 and x n-1 is x d-1. Hint: Use the Euclidean algorithm. 4. If E and F are subﬁelds of GF ( p k ) with p m and p n elements respectively, use the previous exercise to show that E ∩ F contain p d elements, where d = gcd( m,n )....
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