Can a circle of radius $\sqrt[3]{\frac{3}{4}\sqrt{\text{irrational}}}$, or, in other words, a circle of some transcendental radius be constructed? For example, the radius can be $\sqrt[3]{\frac{3}{4}{\pi^2}}$ or $\sqrt[3]{\frac{3}{4}e^2}$. The question can be reframed into "Can we construct a line of transcendental length or any irrational length?" (Here, the term : 'any' implies all possible irrational values). It is not necessary that a straightedge-compass construction is required. Any other way, if possible, would equally be fine.
This is just a random question but I a bit startled though I believe that its not possible. But I don't see a proper reason.
Best Answer
This question comes down to what you mean by 'construct.' If you mean straight-edge and compass, the answer is negative: you can only construct a very special family of irrational numbers, in particular those that lie in some finite tower of degree $2$ field extensions of $\mathbb{Q}$ (link is the same as GEdgar left in a comment). The same goes for any process that comes down to adjoining roots of polynomials: you won't be able to get any transcendental numbers of these extensions. This is just because the transcendental numbers are exactly those that cannot be created from the algebraic properties of $\mathbb{Q}$ alone.
If you allow yourself to use the standard order on $\mathbb{Q}$ and set operations you can construct the real numbers from the rationals via Dedekind cuts. Then you can construct $e$ and the complex plane. With these tools, for any $x \in \mathbb{R}$ you can take the image of $x e^{it}$ and get a circle of radius $x$. An equally valid approach is, once you have $\mathbb{R}$, to construct $\mathbb{R}^2$ and then just take the set of points of standard distance $x$ from the origin. There's no usage of the axiom of choice in these processes, so all steps in each process can be explicitly described and there is a sense in which we can meaningfully call these 'constructions' as well.
In short, if you allow only algebraic tools, you cannot do this. With analytic tools you can.