Prove Trancendence of e^(n/m): Int & Num Solutions

[happyg1]

Ahhhhhh...
A little caffeine and sugar help my brain a lot sometimes.
I Think I understand this now. It's almost trivial, the a^n and a algebraic thing, I'm trying to make it harder than it is.
So now I have that a^(1\m) algebraic implies a algebraic and that a^n algebraic implies a algebraic, so then a^((1\m)^n) algebraic implies a algebraic. I just need to fill in the reasoning for each step and then e^(n\m) CAN'T be algebraic.

Am I finally on target here?
CC

 

[Dick]

Ahhhhhh...
A little caffeine and sugar help my brain a lot sometimes.
I Think I understand this now. It's almost trivial, the a^n and a algebraic thing, I'm trying to make it harder than it is.
So now I have that a^(1\m) algebraic implies a algebraic and that a^n algebraic implies a algebraic, so then a^((1\m)^n) algebraic implies a algebraic. I just need to fill in the reasoning for each step and then e^(n\m) CAN'T be algebraic.

Am I finally on target here?
CC

Yes. Dead on target.

 

[Dick]

Ahhhhhh...
A little caffeine and sugar help my brain a lot sometimes.
I Think I understand this now. It's almost trivial, the a^n and a algebraic thing, I'm trying to make it harder than it is.
So now I have that a^(1\m) algebraic implies a algebraic and that a^n algebraic implies a algebraic, so then a^((1\m)^n) algebraic implies a algebraic. I just need to fill in the reasoning for each step and then e^(n\m) CAN'T be algebraic.

Am I finally on target here?
CC

Except you are still being careless with exponents. Seems to be a big day for that.
a^((1/m)^n) is not equal to (a^(1/m))^n.

 

[happyg1]

Thanks for all of your patience and help on this. I learned A LOT about algebraic numbers from this problem. This stuff is SO COOL! (math nerds everywhere are cheering!)
I look back at my initial try UGGGHHHHH!
Thanks again.
CC

 

[happyg1]

So what if m is negative? Then I get $$\frac{1}{c^{1/m}}$$
Can I just refer this case to the previous case where m is not negative?
CC

 

[Dick]

You'll want to prove if c is algebraic then 1/c is. You can do it pretty directly - just solve a polynomial in c to get an expression for 1/c and argue that it is algebraic. Or if you've proved the algebraic numbers are a field, isn't it immediate?