Find a Basis for `CC` over `RR`.

Reformulate the Given Exercise: The task is to find a set of elements B that forms a basis for the field of complex numbers `CC` when it is viewed as a vector space over the field of real numbers
`RR`. We need to find the set B and the dimension `[CC : RR]`.

General Strategy for a Solution:

Understand the Elements: Recall the definition of a complex number. All elements in `CC` are expressed in a standard form involving real numbers.
Identify the Extension Element: Determine the fundamental element `alpha` that extends the base field `RR` to form `CC` , such that `CC = RR(alpha)`.
Find the Dimension: Determine the degree of the minimal polynomial of `alpha` over `RR` , which gives the dimension `n = [CC : RR]`.
Construct the Basis: The basis B is composed of powers of the extension element: `B = {1 , alpha , ... , alpha^{n-1}}`.

Specific Theory Used:

Complex numbers `CC`: The field of comples numbers consists of all numbers of the form a + bi , where a and b are real numbers (scalars from `RR`) , and i is the imaginary unit.
Field Extension: `CC` can be viewed as the simple field extension of `RR` by the imaginary unit i : `bb{CC = RR(i)}`.
Minimal Polynomial for i over `RR`: The non-zero polynomial `bb{p(x) in RR[x]}` of the lowest degree such that `bb{p(i) = 0}`.
Dimension of Field Extension: The degree of the extension `[CC : RR]` is equal to the degree of the minimal polynomial of i over `RR`: `bb{[CC : RR] = deg(x^2 + 1) = 2}`.
Basis for Field Extension: If `bb{[F(alpha) : F] = n}` , the basis is `bb{{1 , alpha , ... , alpha^{n-1}}}`.

Do the Exercise:

Step 1: Determine the Minimal Polynomial and Dimension:
1. The base field is `bb{F = RR}`. The extension element is `bb{alpha = i}`.
2. The minimal polynomial is `bb{p(x) = x^2 + 1}` , which has degree `bb{n = 2}`.
3. The dimension of the vector space is `bb{[CC : RR] = 2}`.
Step 2: Construct the Basis:
1. Since the dimension `n = 2` , the basis set B must contain two elements.
2. Using the form `B = {1 , alpha}` with `alpha = i` , we get `bb{B = {1 , i}}`.

Verification:

Spanning: Any complex number `z in CC` has the form `bb{z = a + bi}` , where `a , b in RR`. This is a linear combination of the basis elements 1 and i with scalars a and b from the field `RR`: `bb{z = a * 1 + b * i}`.
Linear independence: We must show that `c_1 * 1 + c_2 * i = 0` , where `c_1 , c_2 in RR` , implies `c_1 = 0` and `c_2 = 0`. Let `c_1 + c_2 * i = 0`. Since `c_1` and `c_2` are real numbers , for this equation to hold , both the real part `c_1` , and the imaginary part `c_2` must be zero. Thus , `c_1 = 0` and `c_2 = 0`. So , the set `{1 , i}` is linearly independent.

Conclusion: The basis for `CC` over `RR` is `bb{B = {1 , i}}`. END