# Is pi+e algebraic?

1. Aug 18, 2004

### gravenewworld

Is it algebraic? I remember my professor talking about this probelm, he just swept it under the rug. Are there any proofs that pi+e is algebraic or not?

2. Aug 18, 2004

### jcsd

3. Aug 18, 2004

### matt grime

also at least one of pi+e and pi-e must be transcendental (if they were both algebraic, so would be one half their sum and difference, ie pi and e would be algebraic).

4. Aug 18, 2004

### mathwonk

lets see if wolframs statement is obvious. consider (x-e)(x-pi) = x^2 - (e+pi)x + e*pi.

Now if both e+pi and e*pi were algebraic, then e and pi would be roots of an equation with algebraic coefficients, so wouldn't they both be algebraic?

5. Aug 18, 2004

### shmoe

Yes, and since pi and e are both transcendental, the result follow. That the roots of a polynomial with algebraic coefficients are algebraic follows from the fact that if F2 is and algebraic extension of F1 and F1 is an algebraic extension of F, then F2 is an algebraic extension of F (F2, F1, F fields). Proof should be in most algebra texts and isn't too difficult.