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Webprincipal, J is also non-principal, so the conclusion of the Main Lemma is false in this situation. 2. Factorization of ideals and the ideal class group A partial fix for non-unique factorization of elements in rings of algebraic integers is given by unique factorization of ideals. Theorem 1. Let R be the ring of integers in a quadratic field ... WebLemma 1.2. Any set of the form nZ is an ideal. Proof. Let I = nZ. We have to show I satisfies the three properties in the defini-tion of an ideal. (1) Take arbitrary elements a;b 2 I. We have to show that a+b 2 I. Because a 2 nZ, we know that a = nx for some x 2 Z. Because b 2 nZ, we know that b = ny for some y 2 Z. Then a+b = nx+ny = n(x+y ... how to select layer mask in photoshop
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