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.

Strategy: This exercise aims to solidify your understanding of how quotient rings form field extensions and, more importantly , to introduce the concept that these field extensions can be viewed as vector spaces over their base fields.
3. Vector Space Demonstration: To show K is a vector space over `QQ` , we need to verify the vector space axioms:
Vectors: The vectors in this case will be the elements of `K = QQ[x]"/" langle x^2 -2 rangle`.
Scalars: The scalars will be the elements of `QQ`. We need to show that K is an abelian group under addition and that scalar multiplication (multiplication by elements from `QQ`) satisfies distributive and associative laws , and has an identity element. Since K is already a field , and thus a ring , its additive group properties are inherited and the field multiplication will satisfy the scalar multiplication axioms when restricted to `QQ`.
4. Basis and Dimension: The elements of `K = QQ[x]"/" langle x^2 -2 rangle` can be represented in a particular form. We will find a set of elements that can linearly combine to form any element in K , and are linearly independent over `QQ`. This set will be our basis , and the number of elements in the basis will be the dimension.