## Related questions with answers

Prove that if F is a subfield of a field E, and $c \in E,$ then $\theta: F[x] \rightarrow E$ defined by $\theta(f(x))= f(c)$ is a homomorphism.

Solution

Verified$\mathbb{F}$ is a subfield of $\mathbb{E}$ and $c \in \mathbb{E}$. $\theta: \mathbb{F}[x] \rightarrow \mathbb{E}$ is defined by $\theta(f(x))=f(c)$. This map makes sense because $\theta$ is the evaluation map and $f(c) \in \mathbb{E}$. For $f(x),g(x) \in \mathbb{F}[x]$ we also have $f(x),g(x) \in \mathbb{E}[x]$ and hence by the definitions of addition and multiplication in $\mathbb{E}[x]$

$\begin{align*} \theta((f+g)(x)) &= (f+g)(c) \\ &=f(c)+g(c) \\ &=\theta(f(x))+\theta(g(x)) \end{align*}$

and also

$\begin{align*} \theta((fg)(x)) &= (fg)(c) \\ &=f(c)g(c) \\ &=\theta(f(x))\theta(g(x)) \end{align*}$

Therefore, $\theta$ is a homomorphism.

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