Mixing variables in Mod expressions

[John Cremona <john.cremona@gmail.com>]

I am trying to understand the following.  The answer might well be
something like "if you try to do something stupid then you must expect
a stupid answer"

? Mod(x,x^2-3) + Mod(x,x^2-5)
%1 = 0

-- but I was pretending to be a student trying to find the polynomial
satisfied by sqrt(3)+sqrt(5) and doing what seemed natural.  This
version works:

? Mod(x,x^2-3) + Mod(y,y^2-5)
%2 = Mod(x + Mod(y, y^2 - 5), x^2 - 3)
? a = Mod(x,x^2-3) + Mod(y,y^2-5)
%3 = Mod(x + Mod(y, y^2 - 5), x^2 - 3)
? a^2-8
%4 = Mod(Mod(2*y, y^2 - 5)*x, x^2 - 3)
? (a^2-8)^2
%5 = Mod(Mod(60, y^2 - 5), x^2 - 3)
? (a^2-8)^2-60
%6 = 0

but that is not the point;  result %1 is surely going to confuse people.

John