Finite Fields Flashcards
Lecture 8 (8 cards)
What 4 properties do Groups follow?
- Closure i.e. if you combine any two elements in the group, you will always get another element in the group
- Associativity i.e. doesn’t matter how you group operations on numbers, you still get the same result
- Identity element i.e. there is an element that does nothing when used in an operation e.g. 0 in addition
- Inverses i.e. for every element in the group, there’s another one in the group that undoes it
How is a field different from a group?
It can handle two types of operations being performed on it, rather than just one like a group. Typically, fields will be able to handle multiplication and addition.
What 5 rules do Fields follow?
Fields follow:
- Closure i.e. add two elements together or multiply them, and the result stays in the field
- Associativity i.e. grouping of operations doesn’t matter
- Identity i.e. the result when the operation is applied to a number is the original number e.g. 0 in addition, 1 in multiplication
- Inverses i.e. each element has another element that, when the operation is used, the value is 0 (addition) or 1 (multiplication)
- Commutativity i.e. order doesn’t matter of the numbers
What makes a field a finite field?
Finite fields only exist if they have a limited number of elements
How are finite fields represented in mathematical terms?
A field with 11 elements can be represented as GF(11), whereas a field with 81 elements can be represented as GF(81) or GF(3^4)
How does a prime field differ from standard fields?
A prime field is a finite field with exactly p amount of elements, where p is a prime number.
What is a key factor with prime fields?
All members of a prime field have a multiplicative inverse
What is an extension field?
An extension field is a prime field, but the field exceeds the limit of the set. Example:
- GF(2) = {0, 1}
- GF(2^2) = {0, 1, x + 0, x + 1}
- GF(2^3_ = {0, 1, x + 0, x + 1, x^2 + 0, x^2 + 1, x^2 + x, x^2 + x + 1}
