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  $\mathbb{Q}(\sqrt{m},\sqrt{n})=\mathbb{Q}(\sqrt{m}+\sqrt{n})$:

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

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
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