Finite field is defined as a finite set of numbers and 2 operations +(addition) and * (multiplication) that satisfy the following:

- If
*a*and*b*are in the set,*a*+*b*and*a*b*are in the set. This property is closed. - 0 exists and has the property a+0=a. This is additive identity.
- 1 exists and has the property a*1=a. This is multiplicative identity.
- If
*a*is in the set and –*a*is also in the set, and define the value that makes*a+(-a)*=0. This is additive inverse. - If
*a*is in the set and not 0, a^{-1}is in the set, then a*a^{-1}=1. The is multiplicative inverse.

Example:

set e = {0, 1, 2}. Is set e closed? No, set e is not closed since if you take 1+2=3 and 3 is not in the set. This violates rule number 1 from above.

## Define finite set

A set contain elements, such as 0, 1, 2, 3, 4, … p-1.

Math notation for a set: F_{11}={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

In mathematics, the size of set is also called order. In the above example, the size or order for F is 11. Example of order of 983 is F_{983}={0, 1, 2, … 982}

For our purpose, the fields must have an order that is power of a prime.

Note: Prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole number that can be divided evenly into another number.

## Define Finite Field Addition and Subtraction using Modulo Arithmetics

For F_{19}={0, 1, 2, … 18}

For a+b is closed, a+b must also be an element for F_{19}. Use modulo arithmetic to find the a+b element.

a+b=(a+b)%(order of set) in this case, it is 19. So, 7+8=(7+8)%19=15, which is also in the set and therefore makes a+b a closed finite field. Another example, 11+17=(11+17)%19=9

For additive inverse, for a to be in the set of F, -a is an element in set F if -a=(-a)%(order of set). Example: -9=(-9)%19=10. Which means 9+-F(10)=0 since F(10) is 9 and 9+-9=0.

Field subtraction is a-b=(a-b)%(order of set).

- define what finite field is.
- define finite sets.
- finite field python code.

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