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Example text

Now let’s explore some properties of norms and traces. 15 Suppose α ∈ K . Then N K /Q (α) and TK /Q (α) are both in Q. Proof This simply follows because they are the trace and determinant of a matrix with entries in Q. This is a rather abstract definition of the trace and norm of an element, but we can make it a little more concrete. 16 Write σ1 , . . , σn for the embeddings of K into C. If α ∈ K , then n N K /Q (α) = n σk (α) k=1 and TK /Q (α) = σk (α). k=1 Proof Let g denote the minimal polynomial of α over Q.

Further, b itself cannot be in Z; otherwise A2 − b2 d ∈ / Z. 4 Thus B is an odd integer. 7 Rings of Integers of Number Fields 37 A2 − B 2 d ≡ 0 (mod 4) with A and B odd integers. But the squares of odd numbers are all 1 (mod 4). Thus 1 − d ≡ 0 (mod 4). If d ≡ 1 (mod 4), the second case can arise, and the integers are √ {a + b d | either a, b ∈ Z, or both a and b are halves of odd integers}, a set which is easily seen to be the same as that of the √ statement. On the other hand, if d ̸≡ 1 (mod 4), then the only integers are {a + b d | a, b ∈ Z} as claimed.

N ∈ K such that every element of K can be written as a linear combination x1 α1 + x2 α2 + · · · + xn αn where x1 , . . , xn ∈ Q. 1, a special case of the above, where our basis has a particular form. We can ask exactly analogous questions about the ring of integers Z K . 1. Do there exist elements α1 , . . , αn ∈ Z K such that every element of Z K is of the form x1 α1 + x2 α2 + · · · + xn αn for some xi ∈ Z? 2. 1, for some xi ∈ Z? It will turn out that the first question has a positive answer, but the second does not, in general.

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