Prove that the set of numbers ℚ(√2) = { a + b√2 : a, b ∈ ℚ } is a field.
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Prove that the set of numbers ℚ(√2) = { a + b√2 : a, b ∈ ℚ } is a field. (You may use that ℝ is a field; this makes checking e.g. associativity
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- Label each of the following statements as either true or false. The field of real numbers is complete.14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]
- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.
- 15. (See Exercise .) If and with and in , prove that if and only if in . 14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]