easier proof that a circle can't be squared?

[Richard Duffy]

Here's an obscure point I've never been able to track down in the
literature (though I haven't tried too hard). We know that the
circle-squaring problem from antiquity is unsolvable because:
(1) it amounts to constructing (with compass & straightedge) a
line segment of length pi times the length of some given segment;
(2) the lengths which are constructible from a given segment are
all algebraic over the given length; (3) pi is not algebraic.

But why do we have to use the full strength of the decidedly
non-trivial theorem that pi is transcendental? The constructible
lengths (considering the original "given" segment to have length 1)
are precisely those positive reals obtained by iterating a finite
number of quadratic field extensions starting from the rationals.
So all we need is to prove that *if* pi is algebraic, its degree
is not a power of 2. (Maybe this is no easier than proving the
transcendence of pi.)

I think there's a mention in Dantzig's _Evolution of the Number Concept_
that such a proof had in fact been done, but he gave no reference.
Does anyone know the situation?