# using the primitive element of biquadratic field

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

 $\displaystyle\alpha\;:=\;\sqrt{m}\!+\!\sqrt{n}$ (1)

with rational coefficients.

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

 $\displaystyle\alpha^{3}$ $\displaystyle\;=\;(\sqrt{m})^{3}+3(\sqrt{m})^{2}\sqrt{n}+3\sqrt{m}(\sqrt{n})^{% 2}+(\sqrt{n})^{3}$ $\displaystyle\;=\;m\sqrt{m}+3m\sqrt{n}+3n\sqrt{m}+n\sqrt{n}$ $\displaystyle\;=\;(m+3n)\sqrt{m}+(3m+n)\sqrt{n}$

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 $-2m\!+\!2n$ ($\neq 0$) yields

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

Similarly, we have

 $\displaystyle\sqrt{n}\;=\;\frac{\alpha^{3}\!-\!(m\!+\!3n)\alpha}{2(m\!-\!n)}.$ (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  $\mathbb{Q}(\sqrt{m},\,\sqrt{n})=\mathbb{Q}(\sqrt{m}\!+\!\sqrt{n})$:

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

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

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

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 UsingThePrimitiveElementOfBiquadraticField 2013-03-22 17:54:25 2013-03-22 17:54:25 pahio (2872) pahio (2872) 12 pahio (2872) Application msc 11R16 expressing two square roots with their sum irrational sum of square roots BinomialTheorem