## The Number of Elements in a Finite Field is a Power of a Prime Number

## Problem 726

Let $\F$ be a finite field of characteristic $p$.

Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.

Add to solve laterLet $\F$ be a finite field of characteristic $p$.

Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.

Add to solve later Let $\F_3=\Zmod{3}$ be the finite field of order $3$.

Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$.

**(a)** Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have?

**(b)** Let $ax+b+I$ be a nonzero element of the field $\F_3[x]/(x^2+1)$, where $a, b \in \F_3$. Find the inverse of $ax+b+I$.

**(c)** Recall that the multiplicative group of nonzero elements of a field is a cyclic group.

Confirm that the element $x$ is not a generator of $E^{\times}$, where $E=\F_3[x]/(x^2+1)$ but $x+1$ is a generator.

Add to solve laterLet $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

**(a)** Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.

**(b)** Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.

**(c)** Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.

Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.

For any nonzero element $a\in \F_p$, prove that the polynomial

\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.

(Dummit and Foote “Abstract Algebra” Section 13.5 Exercise #5 on p.551)

Add to solve laterLet $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.

Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.

Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.

(The so-called first standard basis vector in $\F_p^n$.)

Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.

Conclude by induction that

\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]