>>12605199>So if I put the rational numbers on a cartesian coordinate system, with integers spanning x and fractions spanning y, is that a field?You've only decided upon the set of elements your algebra has. Now you also need to decide how you define the rules of an operation that's nominally going to represent 'addition' and rules for an operation that's nominally going to represent 'multiplication' (of course, they both need to be binary operations) and then you can test what properties those rules have.
There are a lot properties you can test, but the basic ones are the five listed in pic related (closure, associativity, identity, invertibility, commutativity).
Depending on how many properties your rules for + or * score, you can see what the algebraic structure that the operation forms over your set of elements in pic related. For example if you prove + has closure, associativity and identity then + forms a Monoid over the set. The most symmetric of these is the Abelian Group which has all five properties.
>is that a field?In order to prove + and * to form a Field over a set you have to prove:
>+ is an abelian group (has all five properties: closure, associativity, identity, invertibility, commutativity)>* is an abelian group (has all five properties: closure, associativity, identity, invertibility, commutativity)>* has left and right distributive property over +.The point of this is that when you prove something is a field, you can write and transform equations by treating the elements of the equation like you usually treat equations with numbers (because numbers form a field the way addition and multiplication of numbers is commonly defined). For example, you will know that (x + yz)(y + x) = xy + xx + yyz + zyx because you know that multiplication is distributive over addition and you know multiplication and addition are commutative and associative.
The whole point is to figure out what the rules for solving equations are.
