Define (= 1+√-3. Show that (+i is algebraic over Q. [Hint: Theorem 4.8.] Theorem 4.8 Let T be a commutative ring, and le

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Define 1 3 Show That I Is Algebraic Over Q Hint Theorem 4 8 Theorem 4 8 Let T Be A Commutative Ring And Le 1 (138.39 KiB) Viewed 43 times

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