Algebraic Number is an Eigenvalue of Matrix with Rational Entries

More from my site

• Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given […]
• 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 […]