Universal Property for Simple Field Extensions

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Theorem

Let $F / K$ be a field extension.

Let $\alpha \in F$ be an algebraic over $K$.

Let $\mu_\alpha$ be the minimal polynomial over $\alpha$ over $K$.


Let $\psi : K \left({\alpha}\right) \to L$ be a homomorphism.

Let $\phi = \psi \restriction_K$.

Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced Homomorphism of Polynomial Forms.


Then $\psi\left({\alpha}\right)$ is a root of $\overline \phi \left({\mu_\alpha}\right)$ in $L$.


Conversely let $L$ be a field.

Let $\phi: K \to L$ be a homomorphism.

Let $\beta$ be a root of $\overline \phi \left({\mu_\alpha}\right)$ in $L$.

Then there exists a unique field homomorphism $\psi : K \left({\alpha}\right) \to L$ extending $\phi$ that sends $\alpha$ to $\beta$.


Proof

Let $\psi: K \left({\alpha}\right) \to L$ be a homomorphism.

Let $\phi = \psi \big|_K$.

Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced Homomorphism of Polynomial Forms.

For any $f = a_0 + \dotsb + a_n X^n \in K \left[{X}\right]$ we have:

\(\displaystyle \psi \left({f \left({\alpha}\right)}\right)\) \(=\) \(\displaystyle \psi \left({a_0 + \cdots + a_n \alpha^n}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({a_0}\right) + \cdots + \phi \left({a_n}\right) \psi \left({\alpha}\right)^n\) by the homomorphism property, and the fact that $\phi = \psi \big\vert_K$
\(\displaystyle \) \(=\) \(\displaystyle \overline \phi \left({f}\right) (\psi\left({\alpha}\right))\)

Since $\mu_\alpha \left({\alpha}\right) = 0$ and a Ring Homomorphism Preserves Zero, the above yields:

$\overline \phi \left({f}\right) \left({\psi \left({\alpha}\right)}\right) = 0$

as required.

$\Box$


Conversely let $L$ be a field.

Let $\phi: K \to L$ be a homomorphism.

Let $\beta$ be a root of $\overline \phi \left({\mu_\alpha}\right)$ in $L$.

By Structure of Simple Algebraic Field Extension, there exists a unique isomorphism:

$\Delta: K \left[{\alpha}\right] \to K \left[{X}\right] / \langle \mu_\alpha \rangle$

such that:

$\Delta \left({\alpha}\right) = X + \left\langle{\mu_\alpha}\right\rangle$

and:

$\Delta \big|_K$ is the identity mapping.

By Universal Property of Polynomial Ring there exists a unique homomorphism:

$\chi: K \left[{X}\right] \to L$

such that:

$\chi \left({X}\right) = \beta$

and:

$\chi \big|_K = \phi$

By Universal Property of Quotient Ring there exists a unique homomorphism:

$\psi: K \left[{X}\right] / \left\langle{\mu_\alpha}\right\rangle \to L$

such that:

$\chi = \psi \circ \pi$


So we have the following diagram:

$\begin{xy}\[email protected][email protected]+1em { K \left[{X}\right] \ar[r]^*{\pi} \[email protected]{-->}[rd]_*{\exists ! \chi} & \dfrac {K \left[{X}\right]} {\left\langle{\mu_\alpha}\right\rangle} \[email protected]{-->}[d]^*{\exists ! \psi} \ar[r]^*{\Delta} & K \left[{\alpha}\right] \\ & L & }\end{xy}$


Now we have:

$\chi \left({X}\right) = \beta$

and:

$\pi \left({X}\right) = X + \left\langle{\mu_\alpha}\right\rangle$

Therefore, because:

$\chi = \psi \circ \pi$

we have:

$\psi \left({X + \left\langle{\mu_\alpha}\right\rangle}\right) = \beta$

Also:

$\Delta^{-1} \left({\alpha}\right) = X + \left\langle{\mu_\alpha}\right\rangle$

so by the above:

$\psi \circ \Delta^{-1}\left({ \alpha }\right) = \beta$
$\psi \circ \Delta^{-1} \big|_K = \psi\big|_K = \phi$
the choice of $\Delta, \psi$ is unique.

$\blacksquare$