How could we tell this element has zero trace? (Galois Theory)

Question

I was reading this question earlier and I was quite confused by Ege's answer when he said 'Moreover, they have zero trace.' I was wondering how could we tell that? So my question is how did we tell that $ \sqrt{a + b \sqrt{c}} + \sqrt{a - b \sqrt{c}} $ has trace $0$?

I am assuming by the word 'trace', we are talking about the sum of all the roots of the minimal polynomial of $ \sqrt{a + b \sqrt{c}} + \sqrt{a - b \sqrt{c}} $ but I was really unsure on how to find its minimal polynomial. So I was wondering is there a slick way of seeing this comment by Ege?

Many thanks in advance!!

Answer 1

You don't need to calculate the minpoly. The conjugates are

• $\sqrt{a + b \sqrt{c}} + \sqrt{a - b \sqrt{c}}$
• $\sqrt{a - b \sqrt{c}} + \sqrt{a + b \sqrt{c}}$
• $- \sqrt{a + b \sqrt{c}} - \sqrt{a - b \sqrt{c}}$
• $- \sqrt{a - b \sqrt{c}} - \sqrt{a + b \sqrt{c}}$

these all cancel so you get 0.