a field L containing $\mathbb{C}$ with $[L:\mathbb{C}]< \infty\ $ then $L=\mathbb{C}$?

Question

I want to show the following: "a field L containing $\mathbb{C}$ with $[L:\mathbb{C}]< \infty\ $ then $L=\mathbb{C}$"

What I know:

1.$\mathbb{C}$ is algebraically closed, so every polynomial in $\mathbb{C}[x]$ has at least one root in $\mathbb{C}$.

2.If $[L:\mathbb{C}]=\infty$, then there are infinity elements of L that are lin. indep on $\mathbb{C}$.

Can anyone help?

Answer 1

If $[L:\mathbf C]<\infty]$, every element of $L$ is algebraic over $\mathbf C$. As $\mathbf C$ is algebraically closed, this means every element of $L$ lies in $\mathbf C$, hence $L\subset\mathbf C$. As the opposite inclusion is true by hypothesis, we conclude that $L=\mathbf C$.

Answer 2

Let $a\in L$. Since $L$ is finite dimensional over $\mathbb{C}$, $a$ is algebraic over $\mathbb{C}$. The minimal polynomial $f(X)$ over $\mathbb{C}$ is irreducible.

What are the irreducible polynomials in $\mathbb{C}[X]$?