- #1

SMA_01

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## 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+b1x

^{1}+b2x

^{2}+b3x

^{3}+...+b(n-1)x

^{n-1}+bnx

^{n}

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

Any help is appreciated.

Thanks.