Question

let f be a field with 76 elements and let k be a subfield of f with 49 elements. then the dimension of f as a vector space over k is

Answer 1

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,
`\( [f:k] = \frac{{\text{{dim}}(f)}}{{\text{{dim}}(k)}} \)`.

Given that f has 76 elements and k has 49 elements, the dimension of f as a vector space over k is:

`\[ [f:k] = \frac{{\text{{dim}}(f)}}{{\text{{dim}}(k)}} = \frac{{76}}{{49}} \]`

However, since dimensions must be integers, we need to find the closest integer. The closest integer to
`\( \frac{{76}}{{49}} \)` is 2.

In other words, the dimension of f as a vector space over k is 2.