Let E be an extension field of a finite field F, where F has q elements. Let ꭤ ∈ E be algebraic over F of degree n. Prove that F(ꭤ) has qn elements.
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Let E be an extension field of a finite field F, where F has q elements. Let ꭤ ∈ E be algebraic over F of degree n. Prove that F(ꭤ) has qn elements.
We have to prove that F(ꭤ) has qn elements.
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- True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field of .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]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.
- [Type here] True or False Label each of the following statements as either true or false. 3. Every integral domain is a field. [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 ab in a field F. Show that x+a and x+b are relatively prime in F[x].
- 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 .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.)14. Prove or disprove that is a field if is a field.