Pries 605A Algebraic Number Theory spring 2016.
Homework 4. Due Fri 2/19.
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1. Let K = Q( −7). Decompose 33 + 11 −7 into irreducible elements of OK .
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2. Let K = Q( −47). Find two factorizations of 54 into irreducible elements of OK .
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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.
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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 .
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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.