in linear algebra, what does this mean?

Question

Z [ sqrt(2)]

Answer 1

This is a Ring
Rings of Z[x] are of the form anX^n+.......+a1X^1+a0
In your case replace x with sqrt of 2.

Hope this helps.

Answer 2

That's the ring of algebraic integers a + b√2,
where a and b are (ordinary) integers.