I would like to know good references for undergraduate students on this topic. I went to wiki page, but it seems most of them are for graduate level. Anyways, one of questions on this topic is the following: Let's suppose we have
$a+b = A$
in which A is algebraic. I suppose we can conclude the following, assuming that the algebraic numbers is a closed field:
If $b$ is algebraic, then so is $a$. In addition, $a$ can be either rational or irrational.
If $b$ is transcedental, then so is $a$.
Are these conclusions and the assumption correct?
Yes. It is a standard fact that the sum of algebraic numbers is algebraic.
In the first case $a = A - b$, so $a$ is algebraic since $A$ and $b$ are by assumption. For rational $a$ take $A = \sqrt{2} + 1$ and $b = \sqrt{2}$, for irrational $a$, take $b = 1$.
In the second case, suppose $a$ is algebraic. Then $A - a = b$ is algebraic. Hence if $b$ is transcendental, then so is $a$.