Is the sum of an algebraic and transcendental complex number transcendental?

I was wondering if the sum of an algebraic and transcendental complex number is transcendental.

I was thinking if a is algebraic, and b is transcendental, then if a+b is algebraic, then a+b-a is also algebraic since algebraic numbers are closed under additive inverses, but b is transcendental.

Is this a correct approach?

• Yes, that's the right idea. – William Stagner Mar 20 '15 at 5:07
• Put another way, given $\Bbb C$, we see that algebraic numbers form a subgroup of $(\Bbb C,+)$, call it $A$. Now $a+b = b + a \in b + A \neq A$, since $b \not\in A$. Since cosets either coincide, or are disjoint, we see $a + b \not\in A$. – David Wheeler Mar 20 '15 at 8:35