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Let F ⊆K be a field extension and let α∈K. Then the following statements are equivalent: (a) α is algebraic over F. (b) F(α) has finite dimension over F
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- 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]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]True or False Label each of the following statements as either true or false. For each in a field , the value is unique, where
- Label each of the following as either true or false. If a set S is not an integral domain, then S is not a field. [Type here][Type here]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.Let be a field. Prove that if is a zero of then is a zero of
- Label each of the following statements as either true or false. Every f(x) in F(x), where F is a field, can be factored.14. Prove or disprove that is a field if is a field.[Type here] True or False Label each of the following statements as either true or false. 2. Every field is an integral domain. [Type here]
- Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]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.Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in