Let us start with exercises that bridges the concepts of ideals , simple extension fields, and introduces vector spaces. Exercise 1: OK , Exercise 2: OK , Exercise 3: OK

Exercise 4:The Extension `QQ(root(3)(2))`:
Let us consider the field extension `QQ(root(3)(2))` where `root(3)(2)` is the real cube of 2.
Part 1: The Basis:
Question 1: What is the set of all elements in `QQ(root(3)(2))`? OK

Question 2: What is the basis for `QQ(root(3)(2))` as a vector space over `QQ` ?
Any element of `QQ(root(3)(2))` is of the form `a + b root(3)(2) + c root (3)(4)` , where `a , b , c in QQ`. We must rewrite it as a linear combination with rational coefficients: `a * 1 + b * root(3)(2) + c * root (3)(4)` , which indicates that the basis is `B = { 1 , root(3)(2) , root (3)(4) }` , as it also spans the vector space.
Are the basis elements linearly independent?
We must look at `c_1 * 1 + c_2 * root (3)(2) + c_3 * root (3)(4) = 0 `. To be linearly independent , all the coefficients `c_1 , c_2 , c_3` must be equal to zero as the only solution. If `c_3 ne 0` we have `c_2 * root (3)(2) + c_3 * root(3)(4) = -c_1`. The left side will always be a real number not `QQ` , because `c_3 ne 0`. Since `-c_1 in QQ` , this is a contradiction. So , `B = { 1 , root(3)(2) , root(3)(4)}` is spanning and linearly independent.
Which means B is our basis for `QQ(root(3)(2))`. END