Question

If $\mathbf{a}\mathbf{\in }\mathbf{F}$, describe the field F [x]/(x - a).

Short answer

F [x]/(x - a) is isomorphic toF.

Step-by-step solution

Statement of Theorem 4.15

Theorem 4.15states consider F as a field, $f\left(x\right)\in F\left[x\right]$ . The remainder isf (a), if f (x)is divided by the polynomial x - a.

Describe the field F [x]/(x - a)

The field F [x]/(x - a) will be isomorphic to F. There is natural inclusion $F\to F\left[x\right]/(x-a)$with$r\mapsto \left[r\right]$; that is an injective homomorphism. According to theorem 4.15, for every $f\left(x\right)\in F\left[x\right]$, we get $\left[f\left(x\right)\right]=\left[f\left(a\right)\right]$in $F\to F\left[x\right]/(x-a)$that will be contained in the copy of F.

It impliesF [x]/(x - a) is isomorphic to F.

Hence,F [x]/(x - a) is isomorphic to F.