How do i show this is transendental

**thegarden** · November 30th 2008, 02:02 AM

If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.

**running-gag** · November 30th 2008, 02:23 AM

Maybe I am wrong but suppose that (e-pi)^3 is not transcental
It means that (e-pi)^3 is solution of a polynomial equation with integer coefficients

$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\;=\;0$

Then

$a_n((e-\pi)^3)^n + a_{n-1}((e-\pi)^3)^{n-1} + ... + a_1(e-\pi)^3 + a_0\;=\; 0$

$a_n(e-\pi)^{3n} + a_{n-1}(e-\pi)^{(3n-3)} + ... + a_1(e-\pi)^3 + a_0\;=\; 0$

Then e-pi is solution of
$a_nx^{3n} + a_{n-1}x^{(3n-3)} + ... + a_1x^3 + a_0\;=\; 0$
Which is not possible because e-pi is supposed transcendental

**thegarden** · November 30th 2008, 02:58 AM

Can anyone check if the above is correct please. Thanks.

**CaptainBlack** · November 30th 2008, 04:42 AM

Originally Posted by thegarden

If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.

If $e-\pi$ is transcendental so is $(e-\pi)^n,\ n \in \mathbb{N}$ . Now put $n=3$ and expand.

CB

**mr fantastic** · May 25th 2009, 12:12 AM

Originally Posted by thegarden

If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.

Thread closed due to OP deleting questions after getting help.

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