Number Theory & Abstract Algebra

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Discussion Overview

The discussion revolves around the differences and relationships between abstract algebra and number theory, particularly in the context of a course titled "Abstract Algebra I & Number Theory." Participants explore how these two areas of mathematics intersect and diverge, with a focus on their applications and foundational concepts.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that abstract algebra focuses on abstract structures like groups, rings, and fields, while number theory deals with whole numbers and their properties, such as divisibility and modular relations.
  • Others argue that number theory utilizes various mathematical fields, including algebra, to solve problems, indicating a strong connection between the two areas.
  • A participant notes that modular arithmetic serves as a common ground between abstract algebra and number theory, highlighting that concepts like remainders mod p can form fields and groups.
  • Some contributions mention specific theorems, such as Fermat's Little Theorem, as applications of group theory within number theory.
  • There is acknowledgment that while abstract algebra can be seen as a tool for number theory, number theory also encompasses topics that do not rely on abstract algebra.
  • Participants express that the course structure and textbook content may blur the lines between the two subjects, leading to confusion about their distinctions.

Areas of Agreement / Disagreement

Participants generally agree that abstract algebra and number theory are related but express differing views on the extent and nature of their relationship. There is no consensus on a definitive distinction between the two fields, and multiple perspectives on their interplay remain present.

Contextual Notes

Some participants highlight that the definitions and applications of concepts in both fields may depend on the specific context or course material, which could lead to varying interpretations.

Who May Find This Useful

Students and educators in mathematics, particularly those interested in abstract algebra and number theory, may find this discussion relevant as it addresses foundational concepts and course-related queries.

jimmypoopins
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I'm currently taking a course, "Abstract Algebra I & Number Theory" and I'm wondering:

what is the difference between abstract algebra and number theory? the two topics seem meshed together.

i tried googling both of them and it doesn't really help. it's hard to tell the differences between the two.

can anyone give me a solid answer?

edit: I'm mostly wondering because we also have a course "Abstract Algebra II," and a course "Topics in Number Theory," both of which require "Abstract Algebra I & Number Theory" as a prerequisite. I'm only required to take one and i'd rather take the one I'm more interested in and better at.
 
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Gosh, I have trouble seeing the similarity. Abstract algebra is the study of abstract groups, rings, fields, and such; it studies properties and how they generalize. They also classify groups and other such creatures.

Number theory is about whole numbers, divisibility, modular relations, and the like. It then uses other fields like algebra (to gain insight into integers considered as a group under addition, or a ring under addition/multiplication mod p, etc.), real analysis (generating functions, sequences, analytic approximations of the discrete), complex analysis (continuations of number-theoretical functions, special functions like zeta), combinatorics, graph theory, etc.
 
yes, 2 very similar topics my little half-wit friend.
I afraid I can't help you further today, I now take my grandmother to doctors.
sorry for my bad english.
love you all, love to your mothers
C to the T to the remBath
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Number Theory and Algebra are related.

Number Theory uses many areas of mathematics to solve problems, as CRGreathouse pointed out. Algebra seems to be one of the most popular. You'll get to learn about Fermat's Little Theorem (Euler's Theorem), which is one of the most common applications from Group Theory to Number Theory.
 
thanks for the replies guys. i had trouble understanding because the course is entitled abstract algebra & number theory, while the book is just "abstract algebra."

ch1 involved division algorithm, random proofs about integers,
ch2 involved modular arithmetic,
and now ch3 is about rings.

i think I'm beginning to understand the difference based on your replies.
 
jimmypoopins said:
ch1 involved division algorithm, random proofs about integers,
ch2 involved modular arithmetic,

Those two chapters contain things you use all the time in Number Theory.
 
Jimmypoopens: I'm currently taking a course, "Abstract Algebra I & Number Theory" and I'm wondering: what is the difference between abstract algebra and number theory?

I understand his question. He has a book on Abstract Algebra that starts with the development of the integers, much of which he might also find in a Number Theory book. I suggest he look beyond the beginning chapters for his answer.
 
I see where he's coming from too. Remainders mod p form a field... remainders mod n in general form a group. Modular arithmetic and Abstract Algebra are essentially the same thing. How about the whole concept of the "algebraic number" too? Abstract Algebra has applications in number theory.

There's plenty of Number Theory that has nothing do do with Abstract Algebra, though. Just wiki Analytic Number Theory and you'll find plenty.
 
rodigee said:
remainders mod n in general form a group.

When the operation is addition, they always do.

When the operation is multiplication, they never do. (You need to take out the number 0 and what else must you do?)
 
  • #10
JasonRox said:
When the operation is addition, they always do.

When the operation is multiplication, they never do. (You need to take out the number 0 and what else must you do?)

Right, sorry for not specifying that. I think you'd need to take remainders mod p in addition to what you noted for the multiplication to be a group.
 
  • #11
abstract algebra is a tool that can be applied to the subject of number theory.

i.e. one is abstract, the other is concrete.
 

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