Example of an Infinite Algebraic Extension
Problem 499
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 $\C$.
Add to solve later Tagged: Eisenstein’s criterion
Add to solve later Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$
Problem 399
Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.
Add to solve later Equation $x_1^2+\cdots +x_k^2=-1$ Doesn’t Have a Solution in Number Field $\Q(\sqrt[3]{2}e^{2\pi i/3})$
Problem 358
Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$.
Add to solve later Application of Field Extension to Linear Combination
Problem 335
Consider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$.
Let $\alpha$ be any real root of $f(x)$.
Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.
Add to solve later Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension
Problem 334
Prove that the polynomial
\[f(x)=x^3+9x+6\]
is irreducible over the field of rational numbers $\Q$.
Let $\theta$ be a root of $f(x)$.
Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.
Add to solve later Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain
Problem 333
Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.
Prove that the polynomial
\[f(x)=x^n-t\]
in the ring $S[x]$ is irreducible in $S[x]$.
Add to solve later Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
Problem 234
Show that the polynomial
\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.
Add to solve later Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group
Problem 231
Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.
Add to solve later $\sqrt[m]{2}$ is an Irrational Number
Problem 179
Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.
Add to solve later Galois Group of the Polynomial $x^p-2$.
Problem 110
Let $p \in \Z$ be a prime number.
Then describe the elements of the Galois group of the polynomial $x^p-2$.
Add to solve later 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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Add to solve later 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 $x^3-\sqrt{2}$ is Irreducible Over the Field $\Q(\sqrt{2})$
Problem 82
Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.
Add to solve later 
