Why does the set of algebraic elements of $K/F$ contain $F$?

Question

Let $K/F$ be a field extension and $\tilde{F}$ the set of elements of $K$ that are algebraic over $F$, i.e.

$$\tilde{F}=\{\alpha \in K \mid \alpha \ \text{algebraic over} \ F \}$$ in my lecture notes for university there is a corollary which states that $\tilde{F}$ is a subfield of $K$ which contains $F$. I'm interested in the last part, i.e. why does $\tilde{F}$ contain $F$? In other words, why is every $x \in F$ algebraic over $F$?

Answer 1

Because $x$ is a root of $T-x\in F[T]$.

Answer 2

Because for all $\;a\in\Bbb F\;,\;\;P_a(x)=x-a\in\Bbb F[x]\;$ is a polynomial with coefficients in $\;\Bbb F\;$ with $\;a\;$ as a root....