# Category: Field Theory

## Determine the Splitting Field of the Polynomial $x^4+x^2+1$ over $\Q$

## Problem 92

Determine the splitting field and its degree over $\Q$ of the polynomial

\[x^4+x^2+1.\]
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## In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.

## Problem 91

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 $\alpha \neq 0$.

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## The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity

## Problem 89

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 later## Algebraic Number is an Eigenvalue of Matrix with Rational Entries

## Problem 88

A complex number $z$ is called * algebraic number* (respectively,

*) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients.*

**algebraic integer**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.

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## Degree of an Irreducible Factor of a Composition of Polynomials

## Problem 83

Let $f(x)$ be an irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be any polynomial in $F[x]$.

Show that the degree of each irreducible factor of the composite polynomial $f(g(x))$ is divisible by $n$.

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## $x^3-\sqrt{2}$ is Irreducible Over the Field $\Q(\sqrt{2})$

## Problem 82

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

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