The field obtained by adjoining $a$ to a prime field

Question

$P$ is a prime field which means that is finite and has p elements, where p is a prime number.

I have problem with understanding the definition of $P(a)$ which is the field obtained by adjoining $a$ to $P$. What does it mean exactly and how does these field look like?

I also have information that $a$ is algebraic over $P$. What does it mean?

Thank you.

Answer 1

1. There is a general concept of adjoining to a field $F$ the root $a$ of an irreducible polynomial (monic) $f \in F[x]$. Then $F(a)$ is a field, which isomorphic as a ring (or as an $F$-algebra) to the quotient ring (algebra) $F[x]/(f)$.
2. It can be shown that, given a prime $p$ and a positive integer $n$, there is a unique (up to isomorphism) field $E$ of order $p^{n}$.
3. It can be shown that $E = P(a)$ for a suitable element $a$.
4. So you are talking of a general finite field.

Answer 2

$a$ is algebraic over $P$ means there is a polynomial $f(x)$ with coefficients in $P$ such that $f(a)=0$. It follows that there is an irreducible polynomial $f(x)$. Then the field $P(a)$ is $P[x]/(f(x))$.