The 'minimal polynomial' of a number z is the f(x) of smallest degree with integer coefficients having z as a root.
For example the minimal polynomial for z = √3 is f(x) = x
2 - 3.
Determine the minimal polynomials of:
A) z = √5 + 3√2
B) z = tan18o.
(In reply to
Part B by looking up z by Jer)
I did it with trig identities. All you need is the multiple angle formula for tan(5x). I'll save myself some typing and refer to https://mathworld.wolfram.com/Multiple-AngleFormulas.html
At the bottom of the article they give recursive formulas to generate higher order multiple angle sums
Let t=tan(x) then tan(5x) = (x^4-10x^2+5)/(5x^4-10x^2+1)
Now plug in z=tan(18deg). Then tan(90deg)=infinity=(z^4-10z^2+5)/(5z^4-10z^2+1).
Getting infinity is essentially dividing by zero; then 5z^4-10z^2+1=0. This is the minimum polynomial of tan(18deg).
re: Part B by looking up z
