Algebraic and Transcendental numbers proof

[SMA_01]

Homework Statement

Let α be a transcendental number and β an algebraic number. Prove that α+β is transcendental.

The Attempt at a Solution

It's known that adding two algebraic numbers results in an algebraic number.

Since β is algebraic, it is a root of a polynomial with integer coefficients.

That is p(x)=Ʃbix^i i=0 to n, p(β)=0.

So,

p(x)= b+b1x¹+b2x²+b3x³+...+b(n-1)x^n-1+bnxⁿ

I'm stuck, I'm not sure where to go from here...

Any help is appreciated.

Thanks.

 

[voko]

By contradiction. Assume the sum is algebraic.

 

[SMA_01]

Okay, assuming the sum is algebraic, that means the root of p(x) added to α will give me a number that can be expressed as a root of another polynomial. How would I proceed?

 

[voko]

You have polynomial R(x) with integer coefficients whose root is α+β. Now consider R(x - β).

 

[SMA_01]

Would x represent any real number? I'm sorry, it's not really clicking with me...

 

[SMA_01]

Are you implying that I should take x=α+β? So that R(x-β) would be R(α+β-β)=R(α)?

 

[voko]

x is the argument of the polynomial R(x - β). What do its coefficients look like?

 

[SMA_01]

R(x-β)=b0+b1(x-β)+b2(x-β)²+...+bn(x-β)ⁿ

But since β is a transcendental number, we can further distribute the bn coefficients right? So for b1(x-β)=b1x-b1β

 

[voko]

β is algebraic according to your original post.

 

[SMA_01]

Oh yeah, sorry I made a mistake. It is algebraic, but I am still trying to figure out what this implies.

 

[voko]

Are the coefficients of R(x - β) integer, rational, algebraic, transcendental? What does that mean with regard to its roots?

 

[SMA_01]

The coefficients are algebraic, and the roots must also be algebraic.

 

[voko]

To prove the statement, you need to consider R(x + β) in the same way. Can you now connect all the pieces?

 

[SMA_01]

So, it R(x-β) and R(x+β) have algebraic coefficients then R(x) mus also have algebraic coefficients right?

 

[voko]

R(x) is defined in #4. Its definition follows from the assumption that α+β is algebraic.

 

[HallsofIvy]

So, it R(x-β) and R(x+β) have algebraic coefficients then R(x) mus also have algebraic coefficients right?

You started by assuming that $\alpha+ \beta$ was algebraic so that it was a root of a polynomial equation $R(x)= 0$ where R is a polynomial with integer coefficients. There is no point is now saying that they must be algebraic!

 

[SMA_01]

I know that, I think I worded what I was trying to say wrong. I meant that algebraic coefficients yield algebraic roots. So R(x) would be a contradiction to this, or am I completely off?

 

[voko]

R(x) does not contradict anything. You assume that $\alpha + \beta$ is algebraic, so there must be a polynomial with integer coefficients it is a root of. This is R(x). Then you consider another polynomial, which is $R(x + \beta)$. $\alpha$ obviously is its root. What does that mean, if $\beta$ is algebraic?

 

[SMA_01]

Do you mean α is a root of R(x+β)? Sorry, I know the solution must be obvious and I'm totally missing it. I'm not really familiar with algebraic and transcendental numbers.

 

[voko]

$P(x) = R(x + \beta)$. What coefficients does P have? What are its roots (algebraic, transcendental)? Is $\alpha$ its root? Why?

 

[SMA_01]

R(x+β)=b0+b1(x+β)+b2(x+β)²+...+bn(x+β)ⁿ

The coefficients are algebraic, and the roots are supposed to be algebraic. α is a root, but it's not algebraic...

 

[voko]

What does this contradiction mean?

 

[SMA_01]

Since α is transcendental, then by definition, it cannot be a root. And then α+β can't be a root either. So it's a contradiction. Is that correct?

As a side question, α is not a root of R(x-β). Why did I need to mention this polynomial then? Is it insufficient to just use R(x+β)?

Thank you

 

[voko]

Yes, that is correct.

I mentioned initially R(x-β) because I hoped that would give you an idea and you would then come up with R(x+β), so you would have more done on your own. R(x-β) is not required for the proof.