How to show $1+\sqrt2 +\cdots+ \sqrt{2^n}$ is algebraic?

Question

How to show that $1+\sqrt2 +\cdots+ \sqrt{2^n}$ is an algebraic?

$x=1+\sqrt2 +\cdots+ \sqrt{2^n}$

$x-1=\sqrt2 +\cdots+ \sqrt{2^n}$

Every other element is an integer so I can move it to the left side. I would like to square both sides, move integer elements to the left and again square both sides and so on. But I have no idea how to write it.

Answer 1

As you noted, every other element is an integer.

Each of the remaining elements can be expressed as an integer times $\sqrt{2}$, hence $$x = a + b\sqrt{2}$$ where $a,b$ are integers.

Can you finish it?

Answer 2

It is algebraic because the sum of algebraic numbers is algebraic and each number $\sqrt{2^k}$ is algebraic.