Find th eminimal polynomial of 2+3 over Q

Question

Jayden-James Duffy · Accepted Answer

To find the smallest degree polynomial f(x) with coefficients in Q, such that

f (\sqrt{2} + \sqrt{3}) = 0 .

The impo
rtant point is that the coefficients should all be in Q. For example,

x - (\sqrt{2} + \sqrt{3})

is not allowed, as its coefficients are not rational numbers.
Now,

\sqrt{3} \notin Q \sqrt{2} .

So, we expect the minimal polynomial of

\sqrt{2} + \sqrt{3}

to have degree

4 (,^{'} \sqrt{2} + \sqrt{3} \in Q (\sqrt{2}) \cdot (\sqrt{3})

, a quadrantic extension of a quadrantic extension of

Q

.
We proceed to find this minimal polynomial by repeated squaring (to clear radicals and obtain rational coefficients)

x = \sqrt{2} + \sqrt{3} \Rightarrow x - \sqrt{2} = \sqrt{3},

squaring

{(x - \sqrt{2})}^{2} = 3 \Rightarrow x^{2} - 2 \sqrt{2} x + 2 = 3

, rearrange

x^{2} - 1 = 2 \sqrt{2} x

, square again

{(x^{2} - 1)}^{2} = 8 x^{2} \Rightarrow x^{4} - 2 x^{2} + 1 = 8 x^{2}

, rearrange
Minimal polynomial is

x^{4} - 10 x^{2} + 1

Find th eminimal polynomial of sqrt2+sqrt3 over QQ