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sol13

# sol13 - Algebra Assignment Solutions Not to be Submitted 1...

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Algebra , Assignment ω Solutions Not to be Submitted 1. With K = Z 2 [ x ] / ( x 3 + x +1) create orthogonal latin squares L 1 = ( a ij ) with a ij = i + j and L 2 = ( b ij ) with b ij = i + xj . Associate K with { 0 , . . . , 7 } by “setting x = 2.” Create a Magic Square of order 8. Solution: First L 1 , the addition table. We write ALL for x 2 + x + 1 so it will fit on the page. 0 1 x x + 1 x 2 x 2 + 1 x 2 + x ALL 1 0 x + 1 x x 2 + 1 x 2 ALL x 2 + x x x + 1 0 1 x 2 + x ALL x 2 x 2 + 1 x + 1 x 1 0 ALL x 2 + x x 2 + 1 x 2 x 2 x 2 + 1 x 2 + x ALL 0 1 x x + 1 x 2 + 1 x 2 ALL x 2 + x 1 0 x + 1 x x 2 + x ALL x 2 x 2 + 1 x x + 1 0 1 ALL x 2 + x x 2 + 1 x 2 x + 1 x 1 0 For L 2 the rows are in a different order. For example, the x 2 row in L 1 (fifth down), becomes the x 3 = x + 1 row (third down) in L 2 . 0 1 x x + 1 x 2 x 2 + 1 x 2 + x ALL x x + 1 0 1 x 2 + x ALL x 2 x 2 + 1 x 2 x 2 + 1 x 2 + x ALL 0 1 x x + 1 x 2 + x ALL x 2 x 2 + 1 x x + 1 0 1 x + 1 x 1 0 ALL x 2 + x x 2 + 1 x 2 1 0 x + 1 x x 2 + 1 x 2 ALL x 2 + x ALL x 2 + x x 2 + 1 x 2 x + 1 x 1 0 x 2 + 1 x 2 ALL x 2 + x 1 0 x + 1 x Translating to 0 , . . . , 7 and putting them together gives

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0 1 2 3 4 5 6 7 1 0 3 2 5 4 7 6 2 3 0 1 6 7 4 5 3 2 1 0 7 6 5 4 4 5 6 7 0 1 2 3 5 4 7 6 1 0 3
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sol13 - Algebra Assignment Solutions Not to be Submitted 1...

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