5. Define ( = 1+√-3. Show that (+i is algebraic over Q. [Hint: Theorem 4.8.] Theorem 4.8 [R. DEDEKIND] Let T be a commut

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5 Define 1 3 Show That I Is Algebraic Over Q Hint Theorem 4 8 Theorem 4 8 R Dedekind Let T Be A Commut 1 (51.18 KiB) Viewed 29 times

5. Define ( = 1+√-3. Show that (+i is algebraic over Q. [Hint: Theorem 4.8.] Theorem 4.8 [R. DEDEKIND] Let T be a commutative ring, and let S be a subring of T. Then IT (S) is a subring of T. PROOF. Let p and q be elements in IT (S), and set A := {p, q). Then, by Proposition 4.6, S[A] ≤ IT(S). Since p, q € S[A] and S[A] is a subring of T, p− q € S[A] and pq € S[A]. Thus, p− q = IT(S)