Question

1. Show that the map$\mathrm{\phi }:\mathrm{F}\left(\mathrm{x}\right)\to \mathrm{F}\left(\mathrm{x}\right)$given byrole="math" localid="1648627349264" $\mathrm{\phi }\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{f}\left(\mathrm{x}+1_{\mathrm{F}}\right)$is an isomorphism such that$\mathrm{\phi }\left(\mathrm{a}\right)=\mathrm{a}$ for every$\mathrm{a}\in \mathrm{F}$ .
2. Use Exercise 31 to show that is role="math" localid="1648627097011" $\mathrm{f}\left(\mathrm{x}\right)$irreducible in $\mathrm{F}\left(\mathrm{x}\right)$if and only if$\mathrm{f}\left(\mathrm{x}+1_{\mathrm{p}}\right)$ is.

Short answer

1. It is proved that the map$\mathrm{\phi }:\mathrm{F}\left[\mathrm{x}\right]\to \mathrm{F}\left[\mathrm{x}\right]$given by$\mathrm{\phi }\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{f}\left(\mathrm{x}+1_{\mathrm{F}}\right)$ is an isomorphism.

Step-by-step solution

Statement of Exercise 22

Exercise 22states consider $R$the commutative ring and $\mathrm{k}\left(\mathrm{x}\right)$a fixed polynomial in$R\left(x\right)$ . Show that there exists a unique homomorphism$\phi :R\left[x\right]\to R\left[x\right]$ in which$\phi \left(r\right)=r$for every $r\in R$and $\phi \left(x\right)=k\left(x\right)$.

 Show that the mapφ:Fx→Fx given byφfx=fx+1F is an isomorphism such thatφa=a for all a∈F

Corollary 4.20states consider $F$an infinite field and $f\left(x\right),g\left(x\right)\in F\left[x\right]$. Then $f\left(x\right)$and $g\left(x\right)$provide the same function from $F$to$F$ if$f\left(x\right)=g\left(x\right)$ is in $F\left[x\right]$.

For the specific case $k\left(x\right)=x-1$, the map $\phi$would be the homomorphism stated inexercise 4.1.22. It is observed that for any $a\in F$, we have that

$$\begin{gathered} \phi \left(a\right)=\phi \left(a{x}^0\right) \\ =a{\left(x+1\right)}^0 \\ =a1 \\ =a \end{gathered}$$

Let the homomorphism in exercise 4.1.22 with $k\left(x\right)=x-1$. $\phi$and $\psi$are inverses of one another, indicatingthat they are bijective,and as a result, $\phi$is an isomorphism.

Hence, it is proved that map$\phi :F\left[x\right]\to F\left[x\right]$ given by$\mathrm{\phi }\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{f}\left(\mathrm{x}+1_{\mathrm{F}}\right)$ is an isomorphism.