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Homeirreducible polynomials obtained from biquadratic fields

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# irreducible polynomials obtained from biquadratic fields

###### Corollary.

Let $m$ and $n$ be distinct squarefree integers, neither of which is equal to $1$. Then the polynomial

$x^{4}-2(m+n)x^{2}+(m-n)^{2}$ |

is irreducible (over $\mathbb{Q}$).

###### Proof.

By the theorem stated in the parent entry, $\sqrt{m}+\sqrt{n}$ is an algebraic number of degree $4$. Thus, a polynomial of degree $4$ that has $\sqrt{m}+\sqrt{n}$ as a root must be irreducible over $\mathbb{Q}$. We set out to construct such a polynomial.

$\begin{array}[]{rl}x&=\sqrt{m}+\sqrt{n}\\ x-\sqrt{m}&=\sqrt{n}\\ (x-\sqrt{m})^{2}&=n\\ x^{2}-2\sqrt{m}\,x+m&=n\\ x^{2}+m-n&=2\sqrt{m}\,x\\ (x^{2}+m-n)^{2}&=4mx^{2}\\ x^{4}+(2m-2n)x^{2}+(m-n)^{2}&=4mx^{2}\\ x^{4}+(2m-2n-4m)x^{2}+(m-n)^{2}&=0\\ x^{4}-2(m+n)x^{2}+(m-n)^{2}&=0\qed\end{array}$

Related:

ExamplesOfMinimalPolynomials, BiquadraticEquation2

Major Section:

Reference

Type of Math Object:

Corollary

## Mathematics Subject Classification

12F05*no label found*12E05

*no label found*11R16

*no label found*

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