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First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help.

I like algebra a lot as a mathematics undergraduate student on his 3rd year.I was wondering what research in algebra means?How does it help other sciences ?what are the applications of all those pure mathematics?Any information relevant to how research in algebra is and how does it help the world to improve.Analysis is more rewarding field maybe.Where the the progress of analysis helps more science fields.

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    $\begingroup$ Algebra does help to improve you so that you can improve the world. As to applications, see coding theory, cryptography, and oil industry. $\endgroup$ Commented Nov 8, 2014 at 12:35
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    $\begingroup$ First of all algebra is a very broad subject..people work in specific topics like group theory,ring theory,field theory.Most of the people do pure mathematics not for the search for real world application but it's own beauty.This types of mathematics gives immense pleasure which is more satisfactory than anything. $\endgroup$
    – Ripan Saha
    Commented Nov 8, 2014 at 13:24
  • $\begingroup$ Search this site: mathoverflow.net. For example, see mathoverflow.net/questions/45802/undergraduate-math-research. $\endgroup$
    – user35603
    Commented Nov 8, 2014 at 13:24

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I am a graduate student, so admittedly I have a limited perspective on this right now, but as someone who is interested in algebra, I'll tell you what I have noticed about "graduate" or "research" level algebra. Also, I don't know much about applications to the "real world" right now, so I won't focus on that, but rather to other areas of pure math.

As noted in the comments, there are many different areas of algebra. I have found that beyond a basic first-year course in algebra, most areas in algebra seem to be designed to have applications to other subjects. For example, commutative algebra was developed to help give algebraic geometry a rigorous foundation, and because of applications to number theory. The representation theory of Lie algebras is of interest because it directly helps to understand Lie groups. Homological algebra was developed because it applies to topology, derived functors were developed in part to give a proper setting to things like sheaf cohomology, etc.

In general many of the definitions and notions studied in modern algebra were designed to have direct applications to other areas of math, especially geometry and topology. In fact a lot of the time you will encounter notions that are directly algebraic abstractions of notions you find in other fields. For this reason I recommend getting a general background in math, especially geometry, if you want to do algebra. To me, the algebra feels a lot more motivated that way.

I decided not to do pure algebra because I felt that if the "reason" people are doing some of this algebra comes from geometry, I wanted to learn the geometry too so I could understand the motivation better. Of course you may not feel the same way and decide to do pure algebra. I recommend looking into some of the major areas of research in algebra and asking around or reading up on their applications. I have mentioned some of the ones that I have found are important.

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    $\begingroup$ By geometry what courses do you mean??And any examples? $\endgroup$
    – Jam
    Commented Nov 8, 2014 at 15:43
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    $\begingroup$ Smooth manifolds, lie groups, algebraic geometry, and algebraic topology. Maybe I'm biased, but I feel that geometry is the most "central" area of modern mathematics, and the biggest motivating factor for algebra. The tools of algebra are extremely powerful when applied to geometry, which is what motivates a lot of the research in algebra, as far as I can see. $\endgroup$
    – Seth
    Commented Nov 8, 2014 at 16:22
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    $\begingroup$ Maybe someone else can answer with a different perspective to round things out. $\endgroup$
    – Seth
    Commented Nov 8, 2014 at 16:30

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