Let F be a field and f(x) E F[x] be of degree
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Q: 1. Explain (write words!) how the graph of h(x) = -ax² compares to the graph of g(x) = ax². 2.…
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Q: Q.1 Find Laplace Transform of the given functions: 1. f(t) = (0, 0≤t<T 1, ≤t<∞0 3. f(t) teat sin(bt)…
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![Let F be a field and f(x) E F[x] be of degree
n> 1. Then the exists a field extension K of F
and a1,..., an E K such that (a) f(x) =
c(x-a1)... (x-an) for some c E F. (b) K=
F(a1,..., an) and dimp(K) ≤n!..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42552fa1-098f-4085-a219-abaefe2c68be%2F07ea5c5f-6441-4baa-9d3c-6218b0f470a4%2F265gnz_processed.jpeg&w=3840&q=75)
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- Prove Theorem If and are relatively prime polynomials over the field and if in , then in .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]Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.
- 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.8. Prove that the characteristic of a field is either 0 or a prime.Let be a field. Prove that if is a zero of then is a zero of
- 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]Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)
- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inSince 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.In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , ,