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I know almost nothing about transcendental numbers, I know the definition of them and maybe few results about them and that is all.

But the question in the title somehow naturally arises when thinking about transcendental numbers.

I think that it is okay to state it once more in the body of the question and not only in the title so here is the question again:

Suppose that $\alpha$ is some transcendental number and that $\beta$ is algebraic number. Is the sum $\alpha + \beta$ always transcendental?

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    $\begingroup$ Yes, as the set $\mathbf A := \{x \in \mathbf C: \text{$x$ is algebraic}\}$ is a subfield of $\mathbf C$. $\endgroup$ – martini Dec 4 '15 at 14:16
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If α + β were algebraic, α = (α + β) – β would be algebraic, since algebraic numbers are a field.

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