[[Algebraic element]]
# Subalgebra generated by an algebraic element

Let $A$ be a [[K-monoid]] over a [[field]] $\mathbb{K}$ and $a \in A$ be an [[algebraic element]] with [[Algebraic element|minimal polynomial]] $m_{a}(x)$.
Then the [[unital subalgebra]] generated by $a$ is[^2008] #m/thm/falg 
$$
\begin{align*}
\mathbb{K}[a] =
\langle a \rangle_{\leq_{\cat{UAsAlg}_{\mathbb{K}}} A} = \{ p(a) : p(x) \in \mathbb{K}[x], \deg p < \deg m_{a} \}
\end{align*}
$$ 
^eq1

and is isomorphic to
$$
\begin{align*}
\langle a \rangle_{\leq_{\cat{UAsAlg}_{\mathbb{K}}} A}  \cong \frac{\mathbb{K}[x]}{\langle m_{a}(x) \rangle_{\trianglelefteq \mathbb{K}[x]} }
\end{align*}
$$
^eq2

> [!check]- Proof
> Let $M = \deg m_{a}$
> First we will show [[#^eq1]].
> First note that the RHS is clearly a [[vector subspace]],
> so it suffices to show that $a^n \in \mathrm{RHS}$ for all $n \in \mathbb{N}_{0}$.
> Applying the [[The polynomial ring over a field is a Euclidean domain|division algorithm for polynomials]]
> $$
> \begin{align*}
> x^n = q(x)m_{a}(x) + r(x)
> \end{align*}
> $$
> where $\deg r < N$.
> But
> $$
> \begin{align*}
> a^n = q(a) m_{a}(a) + r(a) = r(a)
> \end{align*}
> $$
> so $a^n \in \mathrm{RHS}$.
> 
> For the second statement, let $I = \langle m_{a}(x) \rangle_{\trianglelefteq \mathbb{K}[x]}$.
> It follows from above that to every $b \in \langle a \rangle_{\leq a}$ there corresponds a unique $r_{b}(x) \in \mathbb{K}[x]$ with $\deg r_{b} < M$ such that $r_{b}(a) = b$.
> Let $\varphi$ be the map
> $$
> \begin{align*}
> \varphi : \langle a \rangle _{\leq A} &\to \mathbb{K}[x] / I \\
> b &\mapsto r_{b}(x) + I
> \end{align*}
> $$
> Now $\varphi$ is a [[ring homomorphism|ring isomorphism]], since for any $b,c \in \langle a \rangle_{\leq A}$
> $$
> \begin{align*}
> r_{b+c}(x) + I &= r_{c}(x) + r_{b}(x) + I \\
> r_{bc}(x) &= r_{b}(x)r_{c}(x) + I \\
> r_{1}(x) &= 1
> \end{align*}
> $$
> with an inverse by the [[Polynomial ring#evaluation map]] $\eta(a)$. <span class="QED"/>


  [^2008]: Stated without proof in 2008\. [[Sources/@romanAdvancedLinearAlgebra2008|Advanced Linear Algebra]], §18, p. 259.

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