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

Exercise 2:Field extensions as Vector Spaces.
Let F = `QQ` be the field of rational numbers. Consider the polynomial `p(x) = x^2 - 2` `in QQ[x]`.

Problem statement:
Consider the field extension `L = QQ(sqrt 2)` , which is the smallest field containing both the rational numbers and the square root of 2.
1. Demonstrate that L is a vector space over the field of rational numbers `QQ`.
2. Determine the basis for L as a vector space over `QQ`.
3. Calculate the dimension of L over `QQ` , denoted as `[L : QQ ]`.

Strategy and repetition of Vector Spaces:
A vector space (or linear space) V over a field F is a set V equipped with two operations:

1. Vector Addition: For any u , v `bb in V` , there is a unique element `bb {u + v in V}`.
2. Scalar Multiplication: For any c `in F` and u `bb {in V}` , there is a unique element `c bb {u in V}`.

To show that `L = QQ(sqrt 2)` is a vector space over Q , you need to confirm that these properties hold.
Vector Addition: The elements of L are the vectors. is the sum of any two elements in L also in L ? Does it behave tike regular addition (associative , commutative , etc.) ?
Scalar Multiplication: The elements of `QQ` are scalars. If you multiply an element of L by a rational number , is the result still in L ?
Define the elements of L more concretely:
What does an arbitrary element in `QQ(sqrt 2)` look like ? How can you express every element in this field using a minimal set of building blocks from `QQ` ?
Find a basis: Once you have a clear picture of the elements in L , look for a set of elements that can be used to form any other element in L through a linear combination with rational coefficients. A basis must satisfy two conditions: spanning and linear independence.
Spanning: Every element in L can be written as a linear combination of the basis elements.
Linear independence: The only way to write the zero element 0 as a linear combination of the basis elements is with all scalar coefficients equal to zero.
Calculate the dimension: The dimension of a vector space is simply the number of vectors in a basis. Once you have found a basis , just count its elements.