Exercise 4.4.4

Proof. If $n=[L:F]<\infty$, and $\alpha \in L$, then $(1,\alpha ,{\alpha }^2,\cdots ,{\alpha }^n)$ has $n+1$ elements in a space of dimension $n$. Thus there exists $(a_0,\cdots ,a_n)\ne (0,\cdots ,0)$ such that $a_0+a_1\alpha +\cdots +a_n{\alpha }^n=0$. If we write $P={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=0}^n{a}_i{x}^i$, then $P\ne 0$, and $P(\alpha )=0,\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(P)\le n$.

Conclusion: If $n=[L:F]<\infty$, every $\alpha \in L$ is a root of a nonzero polynomial of degree at most $n$. □