· Let F be a field and a be a non-zero element in F. If af(x) is reducible over F, then f (x) € F[x] is Irreducible prime not irreducible unit reducible
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![· Let F be a field and a be a non-zero element in F. If af (x) is reducible over F,
then f (x) E F[x] is
O Irreducible
O prime not irreducible
O unit
O reducible](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F130a7d35-b2af-4849-b2e8-8c97fc516563%2F19a2a6af-a9c7-4466-8fb3-924266064d0b%2Ff9jcv7j_processed.png&w=3840&q=75)
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- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.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]True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in .
- Let where is a field and let . Prove that if is irreducible over , then is irreducible over .Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest lower bound of S. Prove that if F is a complete ordered field and the nonempty subset S has a lower bound in F, then S has a greatest lower bound in F.True or False Label each of the following statements as either true or false. 4. Any polynomial of positive degree over the field has exactly distinct zeros in .
- Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inProve the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.Let be a field. Prove that if is a zero of then is a zero of
- 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.Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .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]