Undergrad Points of a finite projective line

[Lapidus]

I found in Thompson "From Error-Correcting to Sphere Packing and Simple Groups" this on page 131

How do you compute m/n in a finite field?

Take the equivalence class 5 given above. Why does 2/5 and 18/22 give 5?

thanks

 

[andrewkirk]

The finite field being used is $GF(23)$ which is (isomorphic to) the set of integers modulo 23. Because 23 is prime, that is a field.

2/5=5 because $5\times 5=25=2\mod 23$
18/22=5 because $5\times 22=110=18\mod 23$

If you are operating in a finite field of prime order $p$, the multiplicative inverse of $f$ is the smallest positive solution $x$ of the equation $xf=1+kp$ for $k$ any non-negative integer.