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F f /2 Now, supppose we are given n with each fj even. Then qj j = 02 + (qj j )2 and Theorem 59 imply that each of 2, p1 , . . , pr , q1f1 , . . , qsfs can be expressed as a sum of two squares. It suffices, therefore, to prove that for any integers x1 , y1 , x2 , and y2 , there exist integers x3 and y3 satisfying (x21 + y12 )(x22 + y22 ) = x23 + y32 . This easily follows by setting x3 + iy3 = (x1 + iy1 )(x2 + iy2 ) and taking norms. Homework: (1) Suppose n is a positive integer expressed in the form given in Theorem 60 with each fj even.

We show that we can take φ(β) in the definition of a Euclidean domain to be |NQ(√N ) (β)|. Let β and γ be in R − {0}. If β|γ, then N (β) and N (γ) are rational integers with N (β)|N (γ). It easily follows that φ(β) = |N (β)| ≤ |N (γ)| = φ(γ). Considering now more general β and γ be in R − {0}, we show that there are q and √ r in R such that N ). We need only β = qγ + r and either r = 0 or φ(r) < φ(γ). Define δ = β/γ ∈ Q( √ show that there is a q ∈ R for which |N (δ − q)| < 1. Write δ = u + v N where u and v are rational.

Sometimes one can make use of this important feature of ideals in R and obtain rather general number theoretic theorems. Thus, in some sense ideals are indeed ideal. We will establish that unique factorization exists for the ideals in R momentarily, but first we establish some preliminary results. • What do ideals look like in R? An answer to this question is given by our next theorem. We make use of the notion mentioned above. Theorem 66. If I = (0) is an ideal in R, then there exists β1 , β2 , .

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Algebraic number theory (Math 784) by Filaseta M.

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