Using the special subsets of ordered fields and their definitions, Prove that: a. If 0 ≠ p ∈ Q and a ∉ Q, then x+a ∉ Q and ax ∉ Q. (x ∈ Q) b. sqrt(11) is irrational.
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Using the special subsets of ordered fields and their definitions,
Prove that:
a. If 0 ≠ p ∈ Q and a ∉ Q, then x+a ∉ Q and ax ∉ Q. (x ∈ Q)
b. sqrt(11) is irrational.
Step by step
Solved in 3 steps with 2 images
- Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]14. Prove or disprove that is a field if is a field.Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]
- 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field.Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.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+.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]
- If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .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]True or False Label each of the following statements as either true or false. The field of rational numbers is an extension of the integral domain of integers.