1. ## Algebraic number

Hi.

End of semester cram time. Professor snuck in the definition of an algebraic number in our assignment.

I have been able to figure out that

$\displaystyle 6 + \sqrt[4]{2}$

is an algebraic number because it is a root of $\displaystyle (x-6)^4-2$

$\displaystyle \sqrt{2} + \sqrt{-7}$

$\displaystyle (x)^2+7$ and $\displaystyle (x)^2-2$

Then, my mind goes numb on how to put those together.

Thank you.

2. Let $\displaystyle a = \sqrt{2}+\sqrt{7}$. Then $\displaystyle a^2=2+2\sqrt{14}+7=9+2\sqrt{14}$. Thus $\displaystyle (a^2-9)^2=4\times 14 = 56$ and $\displaystyle a$ is a root of $\displaystyle (x^2-9)^2-56$.

3. Originally Posted by Bruno J.
Let $\displaystyle a = \sqrt{2}+\sqrt{7}$. Then $\displaystyle a^2=2+2\sqrt{14}+7=9+2\sqrt{14}$. Thus $\displaystyle (a^2-9)^2=4\times 14 = 56$ and $\displaystyle a$ is a root of $\displaystyle (x^2-9)^2-56$.
Aha! Exactly what I needed: a systematic approach. I can now do the 15 other ones similar to this.

Thank you very much!!!!

4. You are very welcome. Good luck!