Q2.26. Let I (3-√2) be the principal ideal of Z[√2] generated by 3-√2. Define 0 : Z[√2] → Z7 by 0(a+b√2) = [a + 3b]7. 5

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Q2 26 Let I 3 2 Be The Principal Ideal Of Z 2 Generated By 3 2 Define 0 Z 2 Z7 By 0 A B 2 A 3b 7 5 1 (61.28 KiB) Viewed 32 times

Q2.26. Let I (3-√2) be the principal ideal of Z[√2] generated by 3-√2. Define 0 : Z[√2] → Z7 by 0(a+b√2) = [a + 3b]7. 5 (a) Show that 7 € I. ? (b) Show that any element of Z[√2]/I can be written in the form [a], where a € {0, 1, 2, 3, 4, 5, 6}. (c) Show that is a homomorphism. (d) Show that ker 0 = I. (e) Show that Z[√2]/IF. 2