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

1 Answer
George C.
Jul 17, 2015

#f(z) = (z-i)(z-2+sqrt(3))#

#= z^2-(i+2-sqrt(3))z+i(2-sqrt(3))#

If you want rational coefficients then:

#g(z) = (z-i)(z+i)(z-2+sqrt(3))(z-2-sqrt(3))#

#=(z^2+1)(z^2-4z+1)=z^4-4z^3+2z^2-4z+1#

Explanation:

We are dealing with conjugates here.

To get a real number from #i#, multiply it by #+-i#.

To get a rational number from #2-sqrt(3)#, multiply it by #2+sqrt(3)#.

Basically, from the perspective of the rational numbers #QQ#, 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.

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