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By Ash R.B.

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Continuing Problem 3, show that u2 ∈ 5. Verify the conclusions of Problems 3 and 4 when m = 5 and m = 13. Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζn ) of the rationals is formed by adjoining a primitive nth root of unity ζn . In this chapter, we will find an integral basis and calculate the field discriminant. 1 Some Preliminary Calculations The Cyclotomic Polynomial Recall that the cyclotomic polynomial Φn (X) is defined as the product of the terms X −ζ, where ζ ranges over all primitive nth roots of unity in C.

2. Show that the only units of B are ±1. 3. Show that no factor on one side of the above equation is an associate of a factor on the other side, so unique factorization fails. √ 4. √ √ 5. In Z[ −5] and Z −17], the only algebraic integers √ of norm 1 are ±1. Show that this property does not hold for the algebraic integers in Q( −3). 4 Some Arithmetic in Dedekind Domains Unique factorization of ideals in a Dedekind domain permits calculations that are analogous to familiar manipulations involving ordinary integers.

3. A PRACTICAL FACTORIZATION THEOREM 9 (b) Let p be any prime dividing m. Then p divides √ the discriminant, hence p ramifies. Since x2 − m ≡ x2 = xx mod p, we have (p) = (p, m)2 . This takes care of all odd primes, and also p = 2 with m even. (c) Assume p = 2, m odd. (c1) Let m ≡ 3 mod 4. Then 2 divides the √ discriminant D = 4m, so 2 ramifies. We have x2 − m ≡ (x + 1)2 mod 2, so (2) = (2, 1 + m)2 . √ (c2) Let m ≡ 1 mod 8, hence m ≡ 1 mod 4. An integral basis is {1, (1 + m)/2}, and the discriminant is D √ = m.

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