We can construct a field in Sage that extends the rationals by adding in a fourth root of two, Q[42], with the command \texttt{F.<c> = QQ[2\^(1/4)]}. This is a vector space of dimension 4 over the rationals, with a basis that is the first four powers of c=42 (starting with the zero power).
The command F.vector\_space() will return three items in a triple (so be careful how you handle this output to extract what you need). The first part of the output is a vector space over the rationals that is isomorphic to F. The next is a vector space isomorphism (invertible linear transformation) from the provided vector space to the field, while the third is an isomorphism in the opposite direction. These two isomorphisms can then be used like functions. Notice that this is different behavior than for .vector\_space() applied to finite fields. Create non-trivial examples that show that these vector space isomorphisms behave as an isomorphism should. (You will have at least four such examples in a complete solution.)