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.
1. Maximal ideal: To show `I = langle p(x) rangle` is a maximal ideal in `Q[x]` , we will use the property that for a polynomial `f(x) in F[x]` over a field F , an ideal `langle f(x) rangle` is maximal if and only if f(x) is an irreducible polynomial over F. So the first step is to Show `p(x) = x^2 - 2` is irreducible over `QQ`.
2. Quotient Ring as a field: Once we establish that I is maximal , we can directly apply the fundamental theorem of ring theory which states that if R is a commutative ring with unity and M is a maximal ideal of R, then R/M is a field.