/sci/ - Science & Math

>>15002152
>How do I "find" the addition and multiplication tables for a finitie field K with 4 distinct elements {0, 1, a, b}? Is there any method besides guessing?
I'm not sure what you mean by "guessing". The obvious method is very methodical and clear cut.

Since [math]1 + a \notin \{0, 1, a\}[/math], trivially [math]1 + a = b[/math], which is how we'll write it from now on.
Since [math]a^2 \notin \{0, a \}[/math], either [math]a^2 = 1 + a[/math] or [math]a^2 = 1[/math]. If the latter, [math](1 + a)^2 = 1 + a^2 + 2a = 0[/math], which violates field axioms.

Then we're done, basically.