Does one need to know elementary number theory to study Abstract Algebra?

[AdrianZ]

It's been some time that I've been studying abstract algebra and now I'm trying to solve baby Herstein's problems, the thing I have noticed is that many of the exercises are related to number theory in someway and solving them needs a previous knowledge or a background of elementary number theory. Do I need to study naive number theory before I start solving 'Harder' Problems of Herstein? I think Easier and Middle-Level problems can be solved by reading only the book itself, but what about harder problems?

 

[HallsofIvy]

No, I can think of no reason why one should need to know even basic "number theory" in order to study abstract algebra. I do not know of any problems in Herstein that require number theory. Many simple examples in abstract algebra can be given in terms of "modular arithmetic" but I would not consider that to be contained in number theory.

(Number theory itself is a rather limited study- not nearly as much used in other forms of mathematics as abstract algebra.)

 

[obafgkmrns]

@HallsofIvy:

You may be a bit too dismissive of number theory. Gauss, "inventor" of modular arithmetic, would certainly think so:

"The most beautiful theorems of higher arithmetic have this peculiarity, that they are easily discovered by induction, while on the other hand their demonstrations lie in exceeding obscurity and can be ferreted out only by very searching investigations. It is precisely this which gives to higher arithmetic the magic charm which has made it the favorite science of leading mathematicians, not to mention its inexhaustible richness, wherein it so far excels all other parts of mathematics."

 

[micromass]

No, you don't need any knowledge about number theory to tackle Herstein. Everything you need for the exercises will be covered in the book.

 

[blue_raver22]

Yes, having a basic understanding of elementary number theory can be helpful in studying abstract algebra. Many concepts in abstract algebra, such as divisibility, primes, and modular arithmetic, are rooted in number theory. Therefore, having a foundation in number theory can aid in understanding and solving problems in abstract algebra.

However, it is not necessary to have a deep knowledge of number theory before solving harder problems in abstract algebra. The book itself should provide enough information and examples to help you solve these problems. It may be helpful to refer to a number theory textbook or online resources for additional explanations or examples as needed.

Ultimately, the level of understanding of number theory needed for abstract algebra will vary depending on the specific problems and concepts being studied. It is always beneficial to have a well-rounded knowledge of mathematics, so studying number theory can only enhance your understanding of abstract algebra.