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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:

  1. If $b$ is algebraic, then so is $a$. In addition, $a$ can be either rational or irrational.

  2. If $b$ is transcedental, then so is $a$.

Are these conclusions and the assumption correct?

Mr. N
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    Sounds right to me. – saulspatz Jan 13 '21 at 14:14
  • I think you should just take a book on abstract algebra and study that. This has not so much to do with algebraic number theory but is rather just about algebraic numbers - and you'll learn a lot about that if you start studying field extensions. – Qi Zhu Jan 13 '21 at 14:19
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    The first is equivalent to saying that the algebraics are closed under difference, i.e. they comprise a subgroup of $\langle\Bbb C,+\rangle$ by the Subgroup Test.

    The second is its complement - the uniquitous Complementary Sunbgroup Test, well-known in many instances, e.g. even+odd = odd, integer + noninteger = noninteger, etc, as explained in the prior link.

     – Bill Dubuque Jan 13 '21 at 15:58
  • For these types of elementary proofs I suggest looking for expositions of irrational numbers instead of algebraic numbers. A well known book for this is Numbers: Rational and Irrational by Ivan Niven (1961). – Dave L. Renfro Jan 13 '21 at 16:00
  • @DaveL.Renfro Why do you think that would help? What results do you expect to find there that are3 pertinent here? – Bill Dubuque Jan 13 '21 at 16:35
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    @Bill Dubuque: He asked for undergraduate references for "the topic", and based on the questions asked it seemed to me that what was desired was some elementary proof techniques relating to elementary closure properties (your even and odd integers example is a good example of how nothing all that technical about algebraic numbers is involved), and Niven's book has a lot of that. In fact, Niven even discusses different ways that "if ..., then" statements can be worded, which for the level of students he seems to be asking on behalf (and ESL students), even something this basic could be useful. – Dave L. Renfro Jan 13 '21 at 17:59

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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 $a$.

Mummy the turkey
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