How do you write the polynomial function with the least degree and zeroes i, 2 - √3?

Jul 17, 2015

$f \left(z\right) = \left(z - i\right) \left(z - 2 + \sqrt{3}\right)$

$= {z}^{2} - \left(i + 2 - \sqrt{3}\right) z + i \left(2 - \sqrt{3}\right)$

If you want rational coefficients then:

$g \left(z\right) = \left(z - i\right) \left(z + i\right) \left(z - 2 + \sqrt{3}\right) \left(z - 2 - \sqrt{3}\right)$

$= \left({z}^{2} + 1\right) \left({z}^{2} - 4 z + 1\right) = {z}^{4} - 4 {z}^{3} + 2 {z}^{2} - 4 z + 1$

Explanation:

We are dealing with conjugates here.

To get a real number from $i$, multiply it by $\pm i$.

To get a rational number from $2 - \sqrt{3}$, multiply it by $2 + \sqrt{3}$.

Basically, from the perspective of the rational numbers $\mathbb{Q}$, the numbers $i$ and $- i$ are indistinguishable and the numbers $\sqrt{3}$ and $- \sqrt{3}$ (and thus $2 + \sqrt{3}$ and $2 - \sqrt{3}$) are only distinguishable by ordering - not by 'algebraic' properties. That is, if one of these non-rational numbers is a root of a polynomial with rational coefficients, then its conjugate must also be.