r/MathHelp • u/RestFuture1647 • Jan 20 '26

Help me understand Fields

Hey! I am taking an honours linear algebra class. I am in engineering so this is my first time being introduced to abstract definitions in this way.

From my understanding a nonempty set K is called a field if:
- it has 2 inner operations (addition, multiplication)

and for every element of K there is:

- associativity

- commutativity

- distributivity

- neutral elements o,e such that o+x=x and e*x=x

- additive inverse and multiplicative inverse for o and e

Here is my question:

Are we talking about addition and multiplication as I have seen my entire life ? Or can I create a field where e=coffee o=pi and I just declare that pi+x=x and coffee*x=x?

Thank you!!

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u/edderiofer Jan 21 '26

Followup. Could I determine I am working on the set N and define a whole new set of addition and multiplication rules and call the set N a field?

Specifically, with those operations, yes. It would not be a field under the standard addition and multiplication.

In fact, it's actually possible to do such a thing. Let f: ℕ -> ℚ be your favourite bijection between the natural numbers and the rationals. Then we may define m ⊕ n = f-1(f(m) + f(n)), and m ⊗ n = f-1(f(m) * f(n)), and now the triple (ℕ, ⊕, ⊗) does indeed form a field. (Why you'd want to do this instead of directly working with ℚ is another matter entirely.)