Let us start with exercises that bridges the concepts of ideals , simple extension fields, and introduces vector spaces. Exercise 1: OK , Exercise 2: OK

Exercise 3:The Extension `QQ(i)`:
Let us consider the field extension `QQ(i)` , which is the smallest field containing both the both the rational numbers `QQ` and the imaginary unit `i = sqrt {-1}`.

Part 2: The linear Transformation:
Consider the element `alpha = 1 + i` in `QQ(i)`. We define a linear transformation `bb {L_alpha : QQ(i) rarr QQ(i)}` by multiplication by `alpha:bb {L_alpha (x) = alpha x}`. Remember our basis for `QQ(i)` is B = { 1 , i }.

Question 4: Using this basis , represent the linear transformation `L_alpha` as a matrix. OK

Question 5: What is the characteristic polynomial of this matrix ?
Solution question 5: The characteristic polynomial of a square matrix is defined as `p(x) = det(xI - A)` , where I is the identity matrix `[[1 , 0] , [0 , 1]]` , A is the `n times n` matrix and p(x) is the monic polynomial of degree n. In this case A is the `2 times 2` matrix `[[1 , -1] , [1 , 1]]` and p(x) is the monic polynomial of degree 2. The roots of p(x) are called eigen values. In our case we have:

`xI - A = [[x , 0] , [0 , x]] - [[1 , -1] , [1 , 1]] = [[x - 1 , 1] , [-1 , x - 1]]`. And the determinant will be:

`det (xI - A) = (x - 1)(x - 1) - (-1)* 1 = (x^2 - 2x + 1) -(-1) = bb{x^2 - 2x + 2}` , which is our characteristic polynomial. OK