TITLE: Is the perimeter of an ellipse with integer axes irrational?
QUESTION [15 upvotes]: Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 a E(\epsilon)$ where the eccentricity $\epsilon = \sqrt{1-(b/a)^2}$.
Is the following known?

Q. Is the perimeter of an ellipse with integer-length axes $a$ and $b$ always
  irrational? If not, for which $a,b$ is it rational?

           
           $a,b=2,1$. Perimeter is approximately $9.051752335744220113037500073330267105760842885887$.

Answered by paul Monsky in the comments: The perimeter is transcendental!
Established by Theodor Schneider in 1937.

REPLY [10 votes]: Answered by paul Monsky in the comments: The perimeter is transcendental!
Established by Theodor Schneider in 1937:

 

 

Roughly (Those with better knowledge of Deutsch, please correct!):

The value of an elliptic integral of the first or second kind with algebraic
  coefficients and between algebraic limits is transcendental.