>>11039395a field is a set that has some specific rules
the real numbers, the complex numbers, and the rationals are all examples of fields.
there are also some weird ones like Z_5, the integers mod 5, which is {0, 1, 2, 3, 4}.
What does a field need to satisfy?
There needs to be a way to add two elements of a field to get something back in the field, and there needs to be a way to multiply two elements of the field to get something back in the field.
There needs to be an additive identity (which we call 0) and a multiplicative identity (which we call 1) and these cannot be the same element.
Addition and multiplication also need to be associative and commutative.
We also need additive and multiplicative inverses, so for every x in the field there is an element which we call -x such that -x + x = 0. Similarly, for every NONZERO y in the field there is an element y^{-1} such that y^{-1}y = 1.
Finally, multiplication needs to distribute over addition, so x(y + z) = xy + xz.
Most of these are obviously true for R, C, and Q. A lot of them are also easy to show for Z_5. Think about why it's true that every element of Z_5 besides zero has a multiplicative inverse.
Turns out, for any prime p, Z_p is a field, and for any composite number c, Z_c is not a field. Can you find a number with no multiplicative inverse in Z_4?
Why is this important? We can do linear algebra if the elements of our vectors/matrices are in a field. Otherwise, we often can't invert matrices and solve equations. If we remove the condition that every element of a field has to have a multiplicative inverse (and some other conditions as well, like commutativity of multiplication) we get a ring. There are a lot more rings than fields, and just like how you can define a vector space over a field, you can define a module over a ring. So module theory is like linear algebra for rings. It's a lot harder though, since we can't usually take inverses.
