TITLE: Is the fourth root of 2 constructible?
QUESTION [8 upvotes]: Is $2^{1/4}$ (fourth root of two) constructible using only a straight edge and compass? How would you construct it?
I understand that a number is constructible if it can be done in a finite number of step in a field. 
I believe it is constructible because the degree of $Q(2^{\frac{1}{4}}$ over the rationals is $[Q(2^{1/4})): Q] = 4$, which has the form $2^k$. The degree is 4 because a basis for $2^{1/4}$ is $${1, 2^{1/4},2^{1/2}, 8^{1/4}}.$$ Is this reasoning correct?
If it is constructible, do I need to find the four roots of unity and multiple each by $2^{1/4}$? And then how do I proceed from there?

REPLY [2 votes]: Draw a line $AB$ of length $2$, then extend it to $C$ so $AC=3$. Draw a circle with the diameter $AC$, then draw a line perpendicular to $AC$ through $B$. Let it intersect the circle, at both sides of $AC$, at $D$ and $E$.
Then, $BD=BE=\sqrt2$.
Now take $BD$ or $BE$ as the new $AB$ and repeat.
Then the new $BD$ or new $BE$ is $\sqrt[4]{2}$.