## Example of an Infinite Algebraic Extension

## Problem 499

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

Add to solve laterLet $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. 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 $\Q(\theta)$.

Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.

Prove that the polynomial

\[f(x)=x^n-t\]
in the ring $S[x]$ is irreducible in $S[x]$.

Show that the polynomial

\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.

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 laterProve that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.

Add to solve laterLet $p \in \Z$ be a prime number.

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

Add to solve laterDetermine the splitting field and its degree over $\Q$ of the polynomial

\[x^4+x^2+1.\]
Read solution

Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.

Add to solve laterShow that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.

Add to solve later