advertisement

Pries 605A Algebraic Number Theory spring 2016. Homework 4. Due Fri 2/19. √ √ 1. Let K = Q( −7). Decompose 33 + 11 −7 into irreducible elements of OK . √ 2. Let K = Q( −47). Find two factorizations of 54 into irreducible elements of OK . −1 3. Let K be a number field. The different of K is the ideal D √ of OK so that D = {x ∈ K | TL/K (xOK ) ⊂ Z}. Compute the different of K = Q( −2) using the definition. √ 4. For d square-free, let K = Q( d) and let Od be its ring of integers. Recall that the norm N : K ∗ → Q∗ is a multiplicative map. Find all units of Od for every d < 0. 5. Suppose I p ≡ J p in CLK and that p - |CLK |. Show that I ≡ J in CLK . g 6. Fix g > 1. Suppose that √ n is odd, n > 1, and d = n − 1 is square-free. Show that the ideal class group of Q( −d) has an element of order g.