# A somewhat open ended question. What's the fallout of ____- Theorem?

I would like to pose questions occasionally (perhaps I can make a series if this idea is well-received) of the type of the title. In particular I was thinking:

Proposed Question:

What are some of the immediate consequences of the Lindelmann Weierstraß Theorem? I am interested in collecting a list of some of the things that an undergraduate could prove in 4 or 5 lines after learning LW and knowing some pretty basic results about algebraic numbers. Let's assume we know that $\alpha +\beta i$ is algebraic iff $\alpha$ and $\beta$ are algebraic. Then as soon as one learns LW we can show that entire slews of numbers are transcendental. Knowing how powerful a workhorse LW is I am sure there are many things we can see after just a few sentences. I would almost prefer to see them unproven. Just leave the conclusion and statement like : No more than 3 sentences are required. Another user may elect to put comments below asking for clarification of the proof or may elect to update your answer with the proof. So the question:

What is the fallout after achieving the following Lindemann–Weierstrass Theorem (Baker's reformulation)

For algebraic numbers $a_1, \dots a_n$ not all equal to zero and distinct algebraic numbers $\alpha_1, \dots \alpha_n$ we have

$a_1e^{\alpha_1}+a_2e^\alpha_2+ \dots a_n e^\alpha_n \neq 0$?

This question is intended to be a repository for questions that can be quickly defeated by the LW theorem. It is not for a general discussion on the theorem.

End of Proposed Question

Some Cons to asking questions like this: It fails to meet many of the standards of this site. It is open ended. There are multiple correct answers.

Some Pros: It would be nice to have a place to direct users when they ask questions that are the immediate consequence of the ________-Weierstraß theorem (which we can safely say is like $\epsilon>0$ of all mathematics. I swear $\epsilon \approx .02$. Weierstraß ate up everything and has left us only the crumbs... ) . like this one or this one or this one. We can use something like:

"Hi! Welcome to MSE you have asked question that belongs in our "Is an immediate consequence of a Weierstraß theorem."-series. Here's a link to a collection of those. I am going to include an answer to this particular case on that post.

Another pro would be that I think people would enjoy it. But democracy may prove that false. Vote your feels!

• I like this idea in principle, but I think that coming up with such general canonical discussions of consequences of major theorems is something that a textbook does a far better job of addressing. I'm not convinced that we should be reinventing the wheel here.
– user296602
Jul 12 '18 at 15:52
• Beyond the fact that I think that such questions are likely to be overly broad (and, as @T.Bongers suggests, can be better addressed by textbooks), I think that these questions run afoul of the admonishment to ask questions, not start discussions. Specifically, If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here.
– Xander Henderson Mod
Jul 12 '18 at 16:33
• @Xander I am not sure why a question like this would be seen as inviting discussion. In MO there are several very nice big-picture, big-list questions, some of them close in spirit to the example suggested here, that are quite useful to have around. Jul 12 '18 at 18:09
• I think I can make sure that this question won't generate discussion by explicitly stating something along those lines. "This question is intended to be a repository for questions that can be quickly defeated by the LW theorem. It is not for a general discussion on the theorem." A serious toned sentence may have the effect of removing any perceived invitation to discussion. Jul 12 '18 at 19:10
• I agree with the critique that we can do no better than a textbook covering the LW theorem (though I am curious about how different users will interpret the question). I can only respond that MSE is going to end up answering many of these exercises anyway as undergrads come against them stumped. Might as well meet them in an organized fashion. Jul 12 '18 at 19:16
• I'm sure that given any textbook that discusses Lindemann-Weierstrass there exists another textbook that has an application that's not in the first textbook. I think it would be a service to the community to gather applications in a single place on this website. (But maybe not with proofs – we need to be able to assign some problems to our students that they can't just look up on the internet!) Jul 13 '18 at 1:34
• @GerryMyerson. Is that an upvote? I am finding a pretty luke warm response. Jul 13 '18 at 1:36
• Patience, Mason. Patience. Jul 13 '18 at 1:40
• Who is in a rush?!?! Tell me everything you know about LW!!! Jul 13 '18 at 1:44
• The comments (which are not well upvoted) on the bottom of this well received question: math.stackexchange.com/questions/99339/… seem to indicate that this might be a better idea as a community wiki? I can look into whatever that is if we think that's a better option. Jul 22 '18 at 2:41
• Here is a big list: math.leidenuniv.nl/~evertse/numthex9.pdf Jul 24 '18 at 23:25