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It is an instructive exercise to find generators for the principal ideals P25 , 5 Q2 , P35 and Q53 . We finish this section by showing that U (OK ) is an infinite group whenever K is a real quadratic field. We first show that there are arbitrarily large numbers in OK with small norm. 7 Let K be a real quadratic field with discriminant ∆. Let A be any positive number with A > |∆|. Then for each positive number M , there exists β ∈ OK with β > M and |N (β)| < A. 1 to the following set X = {(x, y) : |x| < AM, |y| < 1/M }.

Nn |∆K | where ∆K denotes the√discriminant of K. In particular when K is real quadratic then MK = 12 ∆K and when K is imaginary quadratic then MK = 2 |∆K |. π The following theorem is valid for all number fields, but we state and prove it only for quadratic fields. 2 (Minkowski) Let K be a quadratic field. Each ideal class of K contains an ideal I of OK with N (I) ≤ MK . Proof First take any nonzero ideal J of OK . Let Λ = σ ¯ (J). 1 to Λ and a suitable region X . To define X we split into the cases of K real and K imaginary.

An ∈ Z. The rank of a free abelian group is uniquely determined. Each abelian group G has a subgroup 2G = {u + u : u ∈ G} and when G is free abelian of rank n, then the index |G : 2G| = 2n . We wish to show that each subgroup of a free abelian group G is also a free abelian group. It suffices to prove this for G = Zn . To warm up, consider the case where n = 1. If H is a subgroup of Z then either H = {0} or H contains some positive number. The group {0} is free abelian of rank 0, but if H = {0} let b be the smallest positive integer in H.

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