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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online