hw11 - Math 3320 Problem Set 11 1 1 Make a list of all the...

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Math 3320 Problem Set 11 1 1. Make a list of all the Gaussian primes x + yi with - 7 x 7 and - 7 y 7. (The only actual work here is to Fgure out the primes x + yi with 0 y x 7, then the rest are obtainable from these by symmetry properties.) 2. ±actor the following Gaussian integers into primes in Z [i] : 3 + 5 i , 8 - i , 10 + i , 5 - 12 i , 35 i , - 35 + 120 i , 253 + 204 i . 3. Apply the Euclidean algorithm in Z [i] to the pair of Gaussian integers 17 + 6 i and 5 - 3 i , and use the resulting equations to Fnd Gaussian integers x and y such that ( 17 + 6 i)x + ( 5 - 3 i)y = 1. 4. In this problem we consider Z - 2. To simplify notation, let ω = - 2, so elements of Z [ω] are sums x + with x,y Z and with ω 2 = - 2. We have N(x + yω) = x 2 + 2 y 2 = (x + yω)(x - yω) . (a) Draw the topograph of x 2 + 2 y 2 including all values less than 70 (by symmetry, it su²ces to draw just the upper half of the topograph). Circle the values that are prime (prime in
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hw11 - Math 3320 Problem Set 11 1 1 Make a list of all the...

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