Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$

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• Application of Field Extension to Linear Combination 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 $\Q$.   Proof. We first prove that the polynomial […]
• $x^3-\sqrt{2}$ is Irreducible Over the Field $\Q(\sqrt{2})$ Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.   Hint. Consider the field extensions $\Q(\sqrt{2})$ and $\Q(\sqrt[6]{2})$. Proof. Let $\sqrt[6]{2}$ denote the positive real $6$-th root of of $2$. Then since $x^6-2$ is […]
• Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group 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.   Proof. Put $\alpha=\sqrt{2+\sqrt{2}}$. Then we have $\alpha^2=2+\sqrt{2}$. Taking square of $\alpha^2-2=\sqrt{2}$, we obtain […]
• The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity 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.   Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
• Determine the Splitting Field of the Polynomial $x^4+x^2+1$ over $\Q$ Determine the splitting field and its degree over $\Q$ of the polynomial \[x^4+x^2+1.\] Hint. The polynomial $x^4+x^2+1$ is not irreducible over $\Q$. Proof. Note that we can factor the polynomial as […]
• Example of an Infinite Algebraic Extension 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 $\C$.   Definition (Algebraic Element, Algebraic Extension). Let $F$ be a field and let $E$ be an extension of […]
• Equation $x_1^2+\cdots +x_k^2=-1$ Doesn’t Have a Solution in Number Field $\Q(\sqrt[3]{2}e^{2\pi i/3})$ 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 $k\geq 1$.   Proof. Note that $\alpha= \sqrt[3]{2}e^{2\pi i/3}$ is a root of the polynomial $x^3-2$. The polynomial $x^3-2$ is […]
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