We will examine algebraic numbers () and a way to determine a polynomial with rational coefficients that will annihilate a given algebraic number. In particular, we will focus on algebraic numbers that can be expressed as a finite combination of additions, subtractions, multiplications, divisions, and n-th roots of integers.
In review of algebraic numbers, we define to be an algebraic number if there exists a non-trivial (that is, with at least one non-zero coefficient) polynomial function with rational coefficients with such that is a root. That is:
We then say that is an annihilator of , and is the smallest degree possible for such a polynomial, we will say that is an -th degree algebraic number.
To that end, consider . To find an annihilator for this particular , we can simply use some algebra:
And so we have a polynomial of degree 6. However, simple algebraic manipulations do not always work for any arbitrary algebraic number. For instance, if , this method would not work.
Instead, we must explore another approach, using linear algebra. We will now view as being a vector space over . That is, we will be concerned with vectors as being algebraic numbers, while the scalars are the rational numbers.
Now for an arbitrary with degree , consider the subspace .
As is an -th degree algebraic number, we have for some polynomial function with coefficients . This implies that is a linearly dependent set, and so we take as a basis for . However, we cannot solve for in terms of basis , nor may we necessarily know .
What we are able to do, however, is perform a change of basis. This can be achieved by expanding powers of and keeping track of possible basis vectors. Once we expand up to a particular power of , say , and found possible basis vectors, we can infer a new basis , including 1 as a basis vector.
Back to the case of , experience will inform us that in expanding powers of , we should expect to see powers of , and products thereof. Thus, we can expect the following basis:
Thus, we will have identified 9 new basis vectors after expanded up to in this example. Clearly, these are computations that should be left to a computer algebra system.
Once we have expanded to , we will have identified , as well as determined each of to with respect to . To that end, we can then also express in terms of basis .