Why is $K[\alpha]$ a polynomial ring and the smallest subring containing $K$ and $\alpha$ of $L$?

Question

Let $L/K$ be a field extension. Why is $$K[\alpha]:=\{ \sum^d_{i=0}c_i\alpha^i:c_i\in K:0\leq i\leq d\}=\bigcap_{K\subset M \subset L; \alpha \in M \text{ ring}} M$$ Reading lecture notes on galois theory one defines $K[T]$ as polynomial ring. Later $K[A]$ is defined as the smallest subring containing set $A\subset L$ and $K$. In particular if $A=\{\alpha\}$ notations above do overlap. But for me it is not obvious that they are equal and it is not really explained in the notes.

Answer 1

Indeed, $K[t]$ can mean different things depending on the context.

If $K$ is any field and $t$ is an indeterminate, then $K[t]$ is the polynomial ring over $K$ in one variable, in other words, the free commutative $K$-algebra on one generator.

If $K\subset L$ is a field extension and $t$ is an element of $L$, then $K[t]$ denotes the $K$-subalgebra of $L$ generated by $t$. In general, this is not a free algebra, since $t$ might satisfy polynomial relations. For example, when $K=\mathbb R$, $L=\mathbb C$ and $t=i$, we have $$ \mathbb R[i] = \mathbb C \cong \mathbb R[x]/\langle x^2+1\rangle, $$ where $\mathbb R[i]$ denotes the subalgebra generated by $i$, $\mathbb R[x]$ denotes the polynomial ring (or free algebra) and the isomorphism is given by $i\mapsto x$, since $x^2+1$ is the minimal polynomial of $i$ in the extension $\mathbb R\subset\mathbb C$.

Note that still $$ K[t] = \left\{\, \sum_{i=0}^d c_i t^i \,\middle|\, c_i\in K, d\in\mathbb N\right\} $$ is true in both settings.