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

Exercise:Exploring a field E as a Vector Space.
Let F = `QQ` be the field of rational numbers. Consider the polynomial `p(x) = x^2 - 2` `in QQ[x]`.
1. Show that the ideal `I = langle p(x) rangle` generated by p(x) in `QQ[x]` is a maximal ideal.
2. Deduce that the quotient ring `K = QQ[x]"/" I` is a field. This is a simple extension field of `QQ`.
3. Demonstrate that K is a vector space over `QQ`.
4. Find a basis for K as a vector space over `QQ` and determine its dimension.

Detailed Solution (including explanation of Vector Space theory used): Before we dive into the solution , let us briefly revisit the definition of a Vector Space.

Introduction to Vector Spaces (Theory):
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}`.

Continuing:
Key Concepts related to Vector Spaces:
Subspace:
A non-empty subset W of a vector space V is a subspace if W is itself a vector space under the operations of V. This usually requires checking closure under addition and scalar multiplication.
Linear Combination:
A vector v is a linear combination of vectors `v_1 , ... , v_n` if `v = c_1 v_1 + ... + c_n v_n` for some scalars `c_1 , ... , c_n in F`.
Span:
The set of all linear combinations of a set of vectors S = {`v_1 , ... , v_n`} is called the span of S , denoted span(S).
Linear Independence:
A set of vectors S = {`v_1 , ... , v_n`} is linearly independent if the only way to form the zero vector as a linear combination is when all scalars are zero: `c_1 v_1 + ... + c_n v_n = 0 rArr c_1 = 0 ,  ...  , c_n = 0`.
Basis:
A set of vectors B = {`b_1 , ... , b_n`} is a basis for a vector space V if:
1. B is linearly independent.
2. B spans V. Meaning every vector in V can be written as a linear combination of vectors in B.
Dimension:
The number of vectors in any basis for a vector space is called the dimension of V , denoted `dim_F (V)`. This number is unique for a given space.