Question

Definition A commutative ring is a ring R that satisfies the additional axiom: R9. Commutative Law of Multiplication. For all a, bER Definition A ring with identity is a ring R that satisfies the additional axiom: R10. Existence of Multiplicative Identity. There exists an element 1R E R such that for all aeR a 1R a and R a a Definition An integral domain is a commutative ring R with identity IRメOr that satisfies the additional axiom: R1l. Zero Factor Property. For all a,b E R, if a b OR, then a OR or b-OR Theorem 15.1 Let a, b, e, d be constructible numbers with c0 and d > 0. Then each of a +b, a - b, ab, a/e and vd is a constructible number Drnnt 3. Prove that the set F of all constructible numbers is a subfield of R and hence is a field Hint: To show that F is a sublield of R, you need to show that is a subring that also satisfies axioms R9, R10, d. You can refer to Theorem 15.1 for much of the verification.

Answer 1

Sub oP Rand tlence ield set of al condstpuctibe number theorem 15. eotics. CO eaels?ence or multiplicative identi

consider nce KCIR LKİs k is subfield of R