Algebraic Number is an Eigenvalue of Matrix with Rational Entries

Problem 88

A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients.

Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of a matrix with rational (resp. integer) entries.

$(\implies)$
Suppose that $z$ is algebraic number (resp. algebraic integer). Then $z$ is a root of a monic polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are rational numbers (resp. integers).

Note that the matrix $A$ has rational (resp. integer) entries. Then the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.
Hence $z$ is an eigenvalue of the matrix $A$.

$(\impliedby)$
Suppose that $z$ is an eigenvalue of a matrix $A$ with rational (resp. integer) entries.
Then $z$ is a root of the characteristic polynomial of $A$.

The characteristic polynomial of $A$ is a monic polynomial with rational (resp. integer) coefficients. Thus $z$ is an algebraic number (resp. integer).

Companion Matrix for a Polynomial
Consider a polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Define the matrix
\[A=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & […]

A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer
Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.
Prove that $\alpha$ is an integer.
Proof.
Suppose that $\alpha=\frac{p}{q}$ is a rational number in lowest terms, that is, $p$ and $q$ are relatively prime […]

In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
Show that the matrix $A=\begin{bmatrix}
1 & \alpha\\
0& 1
\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$.
Comment.
Remark that if $A$ is a square […]

Any Automorphism of the Field of Real Numbers Must be the Identity Map
Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.
Proof.
We prove the problem by proving the following sequence of claims.
Let $\phi:\R \to \R$ be an automorphism of the field of real numbers […]

Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic
Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.
Hint.
Note that any homomorphism between fields over $\Q$ fixes $\Q$ pointwise.
Proof.
Assume that there is an isomorphism $\phi:\Q(\sqrt{2}) \to \Q(\sqrt{3})$.
Let […]

Example of an Infinite Algebraic Extension
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 $\C$.
Definition (Algebraic Element, Algebraic Extension).
Let $F$ be a field and let $E$ be an extension of […]

The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity
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.
Hint.
Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]