Re: exp(sqrt(2))

From: Denis Feldmann (denis.feldmann_at_wanadoo.fr)
Date: 09/02/04 

Date: Thu, 2 Sep 2004 18:28:56 +0200

David McAnally wrote:
> "Denis Feldmann" <denis.feldmann@wanadoo.fr> writes:
>
>> Barnaby Finch wrote:
>>> On 9/1/04 7:02 AM, in article ch4kpu$c0k$1@bunyip.cc.uq.edu.au,
>>> "David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote:
>>>
>>>> Thomas Mautsch <mautsch@math.ethz.ch> writes:
>>>>
>>>>> Is exp(sqrt(2)) irrational or even transcendental?
>>>>
>>>> It's transcendental, by the Generalized Lindemann Theorem.
>>>>
>>>> David
>>>>
>>>> -----
>>>
>>> I believe I read that if x is algebraic (and not zero), then e^x is
>>> transcendental. If e^x is algebraic (and not 1), then x is
>>> transcendental. I'm sure it's possible to choose a transcendental x
>>> such that e^x is also transcendental. Is all this the case for all
>>> b^x, where b is a real transcendental?
>
>> Gelfond-Schneider says that if x is algebraic (not 0 or1) and y is
>> algebraic and not rational, then x^y is transcendental.
>
> There is a generalization of this result. Let b_1, ..., b_k be
> nonzero algebraic numbers, and suppose log(b_1), ..., log(b_k), have
> values which are linearly independent over Z, then 1, log(b_1),...,
> log(b_k), are linearly independent over the algebraic numbers for
> these specific values of the logarirthms.
>
> It follows from this statement that
>
> (1) if x is algebraic and nonzero, then exp(x) is transcendental (put
> b_1 = exp(x));
>
> (2) e is transcendental (put x = 1 in (1));
>
> (3) if x is algebraic and not equal to zero or one, then all values of
> log(x) are transcendental (put b_1 = x);
>
> (4) all nonzero logarithms of 1 are transcendental (put b_1 = 1, and
> let log(b_1) be a nonzero logarithm of 1);
>
> (5) pi is transcendental (immediately follows from (4));
>
> (6) the result that you quoted above holds (i.e. if x is algebraic
> and not 0 or 1, and if y is algebraic and not rational, then all
> values of x^y are transcendental) - put b_1 = x and b_2 = x^y:
> since log(b_1) and log(b_2) are linearly independent over Z, but
> linearly dependent over the algebraic numbers, then b_1 and b_2
> can't both be algebraic.
>
>> But b=2^sqrt 2 is then transcendental, and b^sqrt(2)=4...
>
> While this is true, what relevance does it have to the transcendence
> or otherwise of exp(sqrt(2))?

The OP asked : if b is transcendental and x algebraic, is necessarily b^x
trancendental (as when b=e)?
>
> David
>
> -----

Relevant Pages

  • Re: exp(sqrt(2))
    ... >> I believe I read that if x is algebraic (and not zero), ... Let b_1, ..., b_k be nonzero ... linearly independent over the algebraic numbers for these specific values ... what relevance does it have to the transcendence or ...
    (sci.math)
  • Re: Question about the logarithm of 3 to the base 2
    ... Your problem was that you were taking an algebraic function of TWO ... The Generalized Gelfond-Schneider Theorem states that is a_1, ..., a_k, ... log, are linearly independent over the rational numbers, then ... to prove the transcendence of log. ...
    (sci.math)
  • Re: Question about the logarithm of 3 to the base 2
    ... Your problem was that you were taking an algebraic function of TWO ... The Generalized Gelfond-Schneider Theorem states that is a_1, ..., a_k, ... to prove the transcendence of log. ... linearly independent over the rational numbers and 3 is algebraic, ...
    (sci.math)
  • Re: Question about the logarithm of 3 to the base 2
    ... How do you get to the transcendence of log_2from Lindemann's Theorem? ... You can prove the transcendence of log_2from the Gelfond-Schneider ... log), log), ..., log) are linearly independent over the ... --- Christopher Heckman ...
    (sci.math)