# Number Theory & Abstract Algebra

• jimmypoopins
In summary, Number Theory and Abstract Algebra are related topics that study different aspects of mathematics.
jimmypoopins
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.

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.
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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?)

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.

abstract algebra is a tool that can be applied to the subject of number theory.

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

## 1. What is Number Theory?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers. It focuses on studying the patterns and structures of numbers, including prime numbers, integers, and rational numbers.

## 2. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It focuses on abstract concepts and the relationships between them, rather than specific numerical values.

## 3. What are some real-world applications of Number Theory and Abstract Algebra?

Number Theory and Abstract Algebra have numerous real-world applications, including cryptography, coding theory, and computer science. They are also used in physics, engineering, and economics to solve complex problems.

## 4. How is Number Theory related to other branches of mathematics?

Number Theory is closely related to many other branches of mathematics, including algebra, geometry, and calculus. It provides the foundation for many mathematical concepts and has connections to other areas such as analysis and topology.

## 5. Can anyone study Number Theory and Abstract Algebra?

Yes, anyone can study Number Theory and Abstract Algebra. However, a strong foundation in basic mathematics, such as algebra and geometry, is necessary before delving into these more advanced topics. It also requires patience, perseverance, and a love for problem-solving.

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