- #1

Office_Shredder

Staff Emeritus

Science Advisor

Gold Member

- 4,623

- 630

## Homework Statement

Find the minimal polynomial of [tex]\frac{\sqrt{3}}{1+2^{1/3}}[/tex] over Q

we'll call this x

## Homework Equations

I wish I knew some :(

## The Attempt at a Solution

By taking powers of x, I was able to show that the extension Q(x) has degree six (since 2

^{1/3}and sqrt(3) are both independently inside Q(x)) and hence the minimal polynomial has degree six. So then I took a general degree six polynomial, plugged in x, and got six equations in six unknowns. This is less than elegant, and I ended with a system of equations (luckily three of the six unknown coefficients were zero):

255b + 261d + 595f = -171

150b + 222d + 460f = -144

105b + 159d + 375f = -108

solving for b,d,f. An attempt at a numerical solution doesn't convince me this has a rational solution, but I wouldn't be surprised if there was a computational error preceding this. Is there a better way to do this?

Last edited: