By Waclaw Sierpinski

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Rn are fields. R1 ⊕ R2 ⊕ · · · ⊕ Rn , 5. D. E. Rowe, Gauss, Dirichlet and the law of biquadratic reciprocity, The Mathematical Intelligencer 10 (1988), 13–26. This paper gives a discussion of the relationship between Gauss and Dirichlet, mainly concerning their contributions to number theory including their work on biquadratic reciprocity. 6. W. C. Waterhouse, Quadratic polynomials and unique factorization, American Mathematical Monthly 109 (2002), 70–72. 2 is taken from this paper. Biographies 1.

6 Sums and Products of Ideals 21 a = 0 so bd = 1. Thus b is a unit and J = b = D ⊃ J , a contradiction. Hence I is maximal. 6 Sums and Products of Ideals In this section we show how to add and multiply ideals to obtain further ideals. First we define the sum of two ideals. 1 (Sum of ideals) Let I and J be ideals in an integral domain D. The sum of I and J , written I + J , is defined by I + J = {i + j | i ∈ I, j ∈ J }. It is readily checked that I + J is also an ideal and that it is the minimal ideal containing both I and J .

CB609-01 CB609/Alaca & Williams August 7, 2003 17:16 Char Count= 0 Exercises 23 Proof: We show first that P ∩ D is an ideal of D. Let a, b ∈ P ∩ D. Then a, b ∈ P and a, b ∈ D. From the first of these, as P is an ideal, we see that a + b ∈ P. From the second, as D is an integral domain, it is closed under addition so that a + b ∈ D. Hence a + b ∈ P ∩ D. Now suppose that a ∈ P ∩ D and d ∈ D. As d ∈ D, a ∈ P and P is an ideal of D, we deduce that da ∈ P. As d ∈ D, a ∈ D and D being an integral domain is closed under multiplication, we see that da ∈ D.