Difference between revisions of "Algebra"
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| − | + | == Overview == | |
| + | In [[mathematics]], '''algebra''' can denote many things. As a subject, it generally denotes the study of calculations on some set. In high school, this can the study of examining, manipulating, and solving [[equation]]s, [[inequality|inequalities]], and other [[mathematical expression]]s. Algebra revolves around the concept of the [[variable]], an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. | ||
| + | Algebra can be used to solve different types of equations, but algebra is also many other things. | ||
| − | == | + | === Abstract Algebra === |
| − | + | {{main|Abstract algebra}} | |
| − | + | '''Abstract''' (or '''higher''', or '''modern''') '''algebra''' deals (in part) with generalisations of the normal operations seen arithmetic and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in higher algebra. | |
| − | |||
| + | === Elementary Algebra === | ||
| + | {{main|Elementary algebra}} | ||
| + | Algebra can be used to solve equations as simple as <math>3x=9</math> but in some cases so complex that mathematicians have not figured how to solve the particular equation yet. | ||
| − | + | As if to add to the confusion, "[[algebra (structure)|algebra]]" is the name for a certain kind of structure in modern algebra. | |
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| + | Abstract algebra also arguably contains the field of [[number theory]], which has important applications in computer science. (It is commonly claimed that the NSA is the largest employer in the USA of mathematicians, due to the applications of number theory to cryptanalysis.) However, number theory concerns itself with a specific structure (the [[ring]] <math>\mathbb{Z}</math>), whereas algebra in general deals with general classes of structure. Furthermore, number theory interacts more specifically with | ||
| + | certain areas of mathematics (e.g., [[analysis]]) than does algebra in general. Indeed, number theory | ||
| + | is traditionally divided into different branches, the most prominent of which are | ||
| + | [[algebraic number theory]] and [[analytic number theory]]. | ||
| + | == Study Guides == | ||
| − | + | * [[Algebra/Introduction|Introductory topics in algebra]] | |
| − | * [[ | + | * [[Algebra/Intermediate|Intermediate topics in algebra]] |
| − | * [[ | + | * [[Algebra/Olympiad|Olympiad topics in algebra]] |
| − | * [[ | + | * [[Algebra/Advanced topics|More advanced topics in algebra]] |
| − | * [[ | + | |
| − | * [[ | + | == Recommended AoPS books == |
| − | + | * [{{SERVER}}/store/book/intro-algebra Introduction to Algebra ] | |
| − | + | * [{{SERVER}}/store/book/intermediate-algebra Intermediate Algebra] | |
| + | |||
| + | == See Also == | ||
| + | |||
| + | * [[Abstract algebra]] | ||
| + | * [[Elementary algebra]] | ||
| + | |||
| + | {{disambig}} | ||
| + | [[Category:Algebra]] | ||
| + | [[Category:Fields of mathematics]] | ||
| + | {{stub}} | ||
Latest revision as of 08:31, 21 February 2025
Contents
[hide]Overview
In mathematics, algebra can denote many things. As a subject, it generally denotes the study of calculations on some set. In high school, this can the study of examining, manipulating, and solving equations, inequalities, and other mathematical expressions. Algebra revolves around the concept of the variable, an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with algebraic manipulation. Algebra can be used to solve different types of equations, but algebra is also many other things.
Abstract Algebra
- Main article: Abstract algebra
Abstract (or higher, or modern) algebra deals (in part) with generalisations of the normal operations seen arithmetic and high school algebra. Groups, rings, fields, modules, and vector spaces are common objects of study in higher algebra.
Elementary Algebra
- Main article: Elementary algebra
Algebra can be used to solve equations as simple as 
but in some cases so complex that mathematicians have not figured how to solve the particular equation yet.
As if to add to the confusion, "algebra" is the name for a certain kind of structure in modern algebra.
Abstract algebra also arguably contains the field of number theory, which has important applications in computer science. (It is commonly claimed that the NSA is the largest employer in the USA of mathematicians, due to the applications of number theory to cryptanalysis.) However, number theory concerns itself with a specific structure (the ring 
), whereas algebra in general deals with general classes of structure. Furthermore, number theory interacts more specifically with
certain areas of mathematics (e.g., analysis) than does algebra in general. Indeed, number theory
is traditionally divided into different branches, the most prominent of which are
algebraic number theory and analytic number theory.
Study Guides
- Introductory topics in algebra
- Intermediate topics in algebra
- Olympiad topics in algebra
- More advanced topics in algebra
Recommended AoPS books
See Also
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