Tagged: 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 laterAlgebraic 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.
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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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