# Fields

We know examples of fields from elementary math. The rational, real and complex numbers with the usual notions of sum and multiplication are examples of fields (these are not finite though).

A finite field is a set equipped with two operations, which we will call \( + \) and \( * \) . These operations need to have certain properties in order for this to be a field:

- If \( a \) and \( b \) are in the set, then \( c=a+b \) and \( d=a*b \) should also be in the set. This is what is mathematically called a closed set under the operations \( + \), \( * \).
- There is a zero element, \( 0 \), such that \( a+0=a \) for any \( a \) in the set. This element is called the additive identity.
- There is an element, \( 1 \), such that \( 1*a=a \) for any \( a \) in the set. This element is the multiplicative identity.
- If \( a \) is in the set, there is an element \( b \), such that \( a+b=0 \). We call this element the additive inverse and we usually write it as \( -a \).
- If \( a \) is in the set, there is an element \( c \) such that \( a*c=1 \). This element is called the multiplicative inverse and we write is as \( a^{-1} \).