Let F be a field, and p(x) E F[x] an irreducible polynomial of degree od.Prove that every coset of Flr]/(p) can be represented by a unique polyno-mial of degree strictly less than d, and moreover that these are all distinct.Prove that if F has q elements, F[x]/(p) has q elements.has ql elements.

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Asked Apr 18, 2019

Abstract Algebra. Please explain everything in detail.

Let F be a field, and p(x) E F[x] an irreducible polynomial of degree od.
Prove that every coset of Flr]/(p) can be represented by a unique polyno-
mial of degree strictly less than d, and moreover that these are all distinct.
Prove that if F has q elements, F[x]/(p) has q elements.
has ql elements.
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Let F be a field, and p(x) E F[x] an irreducible polynomial of degree od. Prove that every coset of Flr]/(p) can be represented by a unique polyno- mial of degree strictly less than d, and moreover that these are all distinct. Prove that if F has q elements, F[x]/(p) has q elements. has ql elements.

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Step 1

To prove the statements regarding the quotient ring F[x]/(p(x)), under the given conditions

Step 2

The first part of the problem (regarding the degree and the uniqueness of the coset representative) is a consequence of the fact that F[x] is a Euclidean domain with degree as the norm.

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Step 3

We have already proved that every coset can be represented by a polynomial b(x) of degree less than the degree of p)(x). Here is the proof of the uniqueness of b(x) (for each coset). The main point is that for ...

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