Extension fields, and their cardinality and roots

Question

I have no idea how to begin answering this question. My notes do not help.

Let $f(X) = X^3$ + $X + 1 ∈ \mathbb Z_5[X]$ ($\mathbb Z_5$ denotes the integers mod 5). Let $E=\mathbb Z_5[X]/(f(X))$.

1. How do you show that $E$ is a field and compute its cardinality?
2. Let $α := X + (f(X)) ∈ E$ so that $E = \mathbb Z_5(α)$ and $α^3 + α + 1 = 0$. How do you find $a, b, c ∈ \mathbb Z_5$ such that $α^{−1} = a + bα + cα^2$ in $E$?
3. And finally, how do you prove that $f(X)$ has 3 distinct roots in $E$?

What I know: if $F$ is a field, $F[x]$ is the smallest ring containing both $F$ and x, and $F(x)$ is the smallest field containing both $F$ and $x$. If $x$ is a variable, then $F[x]$ is the ring of polynomials in the variable $x$ and $F(x)$ is the field of rational functions in the variable $x$. Can I use these facts in the proof? Do I use Kronecker's Theorem?

Any help is appreciated!

Answer 1

It suffices to show that $f(X)$ is irreducible, since irreducible elements in a PID generate maximal ideals, and the quotient of any ring by a maximal ideal is a field. To show that $f$ is irreducible, one can show that $f$ has no roots in $\mathbb{Z}/5\mathbb{Z}$, since $f$ has degree $3$. To compute the cardinality of $F := \mathbb{Z}/5\mathbb{Z}[X]/\langle f(X) \rangle$, view $F$ as a vector space over the field $\mathbb{Z}/5\mathbb{Z}$; what are the basis elements? Finally, the finite field of order $p^{n}$ is the splitting field of the polynomial $X^{p^{n}} - X$ over $\mathbb{Z}/p\mathbb{Z}$. Since $X^{p^{n}} - X$ is the product of all irreducible polynomials of degree dividing $n$ over $\mathbb{Z}/p\mathbb{Z}$, any irreducible polynomial of degree $n$ over $\mathbb{Z}/p\mathbb{Z}$ splits over the finite field of order $p^{n}$. Apply this with $p = 5, n = 3$ and the polynomial $f$. The roots of $f$ in $\mathbb{F}_{125}$ are distinct because $f'(X) = 1$. (In general, any irreducible polynomial over a perfect field is separable.)

Answer 2

There is a theorem that you should look for in your notes (or elsewhere) that states that for a polynomial $f(x)$ over a field $K$, $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible over $K$. So as Alex stated above, it suffices to show that $f(x)$ is irreducible over $\mathbb{Z}/(5)$. Now in regards to the cardinality of $E$, think about what the residue classes like. It will help to remind yourself what it means for two polynomials to be congruent modulo $f(x)$. The division algorithm for polynomials will come into play as well...