Try the fastest way to create flashcards
Question

# 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.

## Recommended textbook solutions

#### A First Course in Abstract Algebra

7th EditionISBN: 9780201763904John B. Fraleigh
2,398 solutions

#### Modern Algebra: An Introduction

6th EditionISBN: 9780470384435John R. Durbin
1,081 solutions

#### Abstract Algebra

3rd EditionISBN: 9780471433347David S. Dummit, Richard M. Foote
1,960 solutions

#### Contemporary Abstract Algebra

8th EditionISBN: 9781133599708Joseph Gallian
1,976 solutions