Answer :
The dimension of f as a vector space over k is 2.
Here is how:
The dimension of f as a vector space over k is given by the degree of the field extension [f:k]. In this case, [tex]\( [f:k] = \frac{{\text{{dim}}(f)}}{{\text{{dim}}(k)}} \)[/tex].
Given that f has 76 elements and k has 49 elements, the dimension of f as a vector space over k is:
[tex]\[ [f:k] = \frac{{\text{{dim}}(f)}}{{\text{{dim}}(k)}} = \frac{{76}}{{49}} \][/tex]
However, since dimensions must be integers, we need to find the closest integer. The closest integer to [tex]\( \frac{{76}}{{49}} \)[/tex] is 2.
In other words, the dimension of f as a vector space over k is 2.