# Tagged: field

## Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal

## Problem 172

Let $R$ be a commutative ring.

Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.

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## 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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## 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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