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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