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# Algebra

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Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.

## Abstract algebra

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We just use "Abstract algebra" as a synonym for algebra.

## Algebraic structure

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A set plus any number of functions , such that each satisfies some properties of choice.
Key examples:
• group: one function
• field: two functions
• ring: also two functions, but with less restrictive properties

### Identity element

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#### Inverse element

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Some specific examples:

### Order (algebra)

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The order of a algebraic structure is just its cardinality.
Sometimes, especially in the case of structures with an infinite number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the degree.

#### Degree (algebra)

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The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.
This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.
Examples:
• the dihedral group of degree n acts on n elements, and has order 2n
• the parameter that characterizes the size of the general linear group is called the degree of that group, i.e. the dimension of the underlying matrices

#### Finite algebraic structure

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Examples:

## Linear algebra

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## Group

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