## 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 laterProblems in Field Theory

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 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 $F$.

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterFind an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.

Add to solve later Let $\zeta_8$ be a primitive $8$-th root of unity.

Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.

Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.

Prove that $\alpha$ is an integer.

Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Add to solve laterLet $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$.

The subfield $\Q(\zeta+\zeta^{-1})$ is called **maximal real subfield**.

Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$.

Add to solve laterConsider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$.

Let $\alpha$ be any real root of $f(x)$.

Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.

Prove that the polynomial

\[f(x)=x^3+9x+6\]
is irreducible over the field of rational numbers $\Q$.

Let $\theta$ be a root of $f(x)$.

Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.

**(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 $\F_3$.

Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.

Add to solve laterLet $\Q$ be the field of rational numbers.

**(a)** Is the polynomial $f(x)=x^2-2$ separable over $\Q$?

**(b)** Find the Galois group of $f(x)$ over $\Q$.

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 laterShow that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal.

Read solution

Let $p \in \Z$ be a prime number.

Then describe the elements of the Galois group of the polynomial $x^p-2$.

Add to solve laterProve that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.

Add to solve later