2.1 Homomorphisms from simple extensions.

Let $F$ be a field, and let $E$ and $E^{\prime}$ be fields containing $F$ . Recall that an $F$ -homomorphism is a homomorphism $\varphi\colon E\rightarrow E^{\prime}$ such that $\varphi(a)=a$ for all $a\in F$ . Thus an $F$ -homomorphism $\varphi$ maps a polynomial

\sum a_{i_{1}\cdots i_{m}}\alpha_{1}^{i_{1}}\cdots\alpha_{m}^{i_{m}},\quad a_{% i_{1}\cdots i_{m}}\in F,\quad\alpha_{i}\in E,

\sum a_{i_{1}\cdots i_{m}}\varphi(\alpha_{1})^{i_{1}}\cdots\varphi(\alpha_{m})% ^{i_{m}}.

An $F$ -isomorphism is a bijective $F$ -homomorphism.

An $F$ -homomorphism $E\rightarrow E^{\prime}$ of fields is, in particular, an injective $F$ -linear map of $F$ -vector spaces, and so it is an $F$ -isomorphism if $E$ and $E^{\prime}$ have the same finite degree over $F$ .

Proposition 2.1

Let $F(\alpha)$ be a simple extension of $F$ and $\Omega$ a second extension of $F$ .

(a)

Let $\alpha$ be transcendental over $F$ . For every $F$ -homomorphism $\varphi\colon F(\alpha)\rightarrow\Omega$ , $\varphi(\alpha)$ is transcendental over $F$ , and the map $\varphi\mapsto\varphi(\alpha)$ defines a one-to-one correspondence

$\{F\text{-homomorphisms }F(\alpha)\rightarrow\Omega\}\leftrightarrow\{\text{% elements of }\Omega\text{ transcendental over }F\}.$
(b)

Let $\alpha$ be algebraic over $F$ with minimal polynomial $f(X)$ . For every $F$ -homomorphism $\varphi\colon F[\alpha]\rightarrow\Omega$ , $\varphi(\alpha)$ is a root of $f(X)$ in $\Omega$ , and the map $\varphi\mapsto\varphi(\alpha)$ defines a one-to-one correspondence

$\{F\text{-homomorphisms }\varphi\colon F[\alpha]\rightarrow\Omega\}% \leftrightarrow\{\text{roots of{\ }}f\text{\ in }\Omega\}.$

In particular, the number of such maps is the number of distinct roots of $f$ in $\Omega$ .

Proof.

(a) To say that $\alpha$ is transcendental over $F$ means that $F[\alpha]$ is isomorphic to the polynomial ring in the symbol $\alpha$ . Therefore, for every $\gamma\in\Omega$ , there is a unique $F$ -homomorphism $\varphi\colon F[\alpha]\rightarrow\Omega$ such that $\varphi(\alpha)=\gamma$ (see 1.5). This $\varphi$ extends (uniquely) to the field of fractions $F(\alpha)$ of $F[\alpha]$ if and only if nonzero elements of $F[\alpha]$ are sent to nonzero elements of $\Omega$ , which is the case if and only if $\gamma$ is transcendental over $F$ . Thus we see that there are one-to-one correspondences between (a) the $F$ -homomorphisms $F(\alpha)\rightarrow\Omega$ , (b) the $F$ -homomorphisms $\varphi\colon F[\alpha]\rightarrow\Omega$ such that $\varphi(\alpha)$ is transcendental, (c) the transcendental elements of $\Omega$ .

(b) Let $f(X)=\sum a_{i}X^{i}$ , and consider an $F$ -homomorphism $\varphi\colon F[\alpha]\rightarrow\Omega$ . On applying $\varphi$ to the equality $\sum a_{i}\alpha^{i}=0$ , we obtain the equality $\sum a_{i}\varphi(\alpha)^{i}=0$ , which shows that $\varphi(\alpha)$ is a root of $f(X)$ in $\Omega$ . Conversely, if $\gamma\in\Omega$ is a root of $f(X)$ , then the map $F[X]\rightarrow\Omega$ , $g(X)\mapsto g(\gamma)$ , factors through $F[X]/(f(X))$ . When composed with the inverse of the canonical isomorphism $F[X]/(f(X))\rightarrow F[\alpha]$ , this becomes a homomorphism $F[\alpha]\rightarrow\Omega$ sending $\alpha$ to $\gamma$ . _□

We shall need a slight generalization of this result.

Proposition 2.2

Let $F(\alpha)$ be a simple extension of $F$ and $\varphi_{0}\colon F\rightarrow\Omega$ a homomorphism from $F$ into a second field $\Omega$ .

(a)

If $\alpha$ is transcendental over $F$ , then the map $\varphi\mapsto\varphi(\alpha)$ defines a one-to-one correspondence

$\{\text{extensions }\varphi\colon F(\alpha)\rightarrow\Omega\text{{\ of }}\varphi_{0}\}\leftrightarrow\{\text{elements of }\Omega\text{{\ % transcendental over }}\varphi_{0}(F)\}.$
(b)

If $\alpha$ is algebraic over $F$ , with minimal polynomial $f(X)$ , then the map $\varphi\mapsto\varphi(\alpha)$ defines a one-to-one correspondence

$\{\text{extensions }\varphi\colon F[\alpha]\rightarrow\Omega\text{{\ of }}\varphi_{0}\}\leftrightarrow\{\text{{roots of }}\varphi_{0}f\text{ in }\Omega\}.$

In particular, the number of such maps is the number of distinct roots of $\varphi_{0}f$ in $\Omega$ .

By $\varphi_{0}f$ we mean the polynomial obtained by applying $\varphi_{0}$ to the coefficients of $f$ . By an extension of $\varphi_{0}$ to $F(\alpha)$ we mean a homomorphism $\varphi\colon F(\alpha)\rightarrow\Omega$ whose restriction to $F$ is $\varphi_{0}$ . The proof of the proposition is essentially the same as that of the preceding proposition (indeed, it is essentially the same proposition).