# Finding the minimal polynomial of an irrational over Q

1. Oct 28, 2016

### PsychonautQQ

1. The problem statement, all variables and given/known data
Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q.

2. Relevant equations

3. The attempt at a solution
Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that +/-(3)^1/2 will be roots of the minimal polynomial, as Q(a):Q = [Q(a):Q((3)^1/2)]*[Q(3^1/2):Q]. I do not know where to go from here, any advice PF?

2. Oct 28, 2016

### lurflurf

$$a=\sqrt{1+\sqrt{3}}$$
so your on the right track looking for 4 conjugates
consider all sign variations of square roots
$$\pm\sqrt{1\pm\sqrt{3}}$$
the minimum polynomial will be
(x-a)(x-b)(x-c)(x-d)
where a,b,c,d are the four conjugates
$$a=\sqrt{1+\sqrt{3}}\\ b=\sqrt{1-\sqrt{3}}\\ c=-\sqrt{1-\sqrt{3}}\\ d=-\sqrt{1+\sqrt{3}}\\$$
another possibly easier approach is to isolate 3 in your equation for a
the minimal polynomial of 3 is
x-3
so
f(a)-3
is the minimal polynomial of a if f(a) is 3 in terms of a