# using the primitive element of biquadratic field

Let $m$ and $n$ be two distinct squarefree^{} integers $\ne 1$. We want to express their square roots as polynomials of

$\alpha :=\sqrt{m}+\sqrt{n}$ | (1) |

with rational coefficients.

If $\alpha $ is cubed (http://planetmath.org/CubeOfANumber), the result no terms with $\sqrt{mn}$:

${\alpha}^{3}$ | $\mathrm{\hspace{0.33em}}={(\sqrt{m})}^{3}+3{(\sqrt{m})}^{2}\sqrt{n}+3\sqrt{m}{(\sqrt{n})}^{2}+{(\sqrt{n})}^{3}$ | ||

$\mathrm{\hspace{0.33em}}=m\sqrt{m}+3m\sqrt{n}+3n\sqrt{m}+n\sqrt{n}$ | |||

$\mathrm{\hspace{0.33em}}=(m+3n)\sqrt{m}+(3m+n)\sqrt{n}$ |

Thus, if we subtract from this the product $(3m+n)\alpha $, the $\sqrt{n}$ term vanishes:

$${\alpha}^{3}-(3m+n)\alpha =(-2m+2n)\sqrt{m}$$ |

Dividing this equation by $-2m+2n$ ($\ne 0$) yields

$\sqrt{m}={\displaystyle \frac{{\alpha}^{3}-(3m+n)\alpha}{2(-m+n)}}.$ | (2) |

Similarly, we have

$\sqrt{n}={\displaystyle \frac{{\alpha}^{3}-(m+3n)\alpha}{2(m-n)}}.$ | (3) |

The (2) and (3) may be interpreted as such polynomials as intended.

Multiplying the equations (2) and (3) we obtain a corresponding for the square root of $mn$ which also lies in the quartic field $\mathbb{Q}(\sqrt{m},\sqrt{n})=\mathbb{Q}(\sqrt{m}+\sqrt{n})$:

$\sqrt{mn}={\displaystyle \frac{{\alpha}^{6}-4(m+n){\alpha}^{4}+(3{m}^{2}+10mn+3{n}^{2}){\alpha}^{2}}{4(-{m}^{2}+2mn-{n}^{2})}}$ |

For example, in the special case $m:=2,n:=3$ we have

$$\sqrt{2}=\frac{{\alpha}^{3}-9\alpha}{2},\sqrt{3}=-\frac{{\alpha}^{3}-11\alpha}{2},\sqrt{6}=\frac{-{\alpha}^{6}+20{\alpha}^{4}-99{\alpha}^{2}}{4}.$$ |

Remark. The sum (1) of two square roots of positive squarefree integers is always irrational, since in the contrary case, the equation (3) would say that $\sqrt{n}$ would be rational; this has been proven impossible here (http://planetmath.org/SquareRootOf2IsIrrationalProof).

Title | using the primitive element of biquadratic field |
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Canonical name | UsingThePrimitiveElementOfBiquadraticField |

Date of creation | 2013-03-22 17:54:25 |

Last modified on | 2013-03-22 17:54:25 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Application |

Classification | msc 11R16 |

Synonym | expressing two square roots with their sum |

Synonym | irrational sum of square roots |

Related topic | BinomialTheorem |