How to Remember the 10 Field Axioms

There are a total of 10 Axioms for Field, it can be quite a challenge to remember all 10 of them offhand.

I created a mnemonic “ACIDI” to remember the 10 axioms. Unfortunately it is not a real word, but is close to the word “acidic”. A picture to remember is “acidic field”, a grass field polluted by acid rain?! 😛

A: Associativity
C: Commutativity
I: Identity
D: Distributivity
I: Inverse

Each of the properties has two parts – Addition and Multiplication. This table from Wolfram summarizes it perfectly:

name addition multiplication
associativity (a+b)+c=a+(b+c) (ab)c=a(bc)
commutativity a+b=b+a ab=ba
distributivity a(b+c)=ab+ac (a+b)c=ac+bc
identity a+0=a=0+a a·1=a=1·a
inverses a+(-a)=0=(-a)+a aa^(-1)=1=a^(-1)a if a!=0

Author: mathtuition88

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