Algebra of polynomials

Section A.1 Algebra of polynomials

In this section we collect, without proofs, some observations and results related to polynomials in one variable over a field. We use these results to deduce some consequences to eigenvalues of a linear map. Throughout this section we assume that \(F\) is a field of characteristic \(0\text{,}\) i.e., every \(n\in\mathbb{N}\) is invertible in \(F\text{.}\) Assume that \(F[t]\) is the collection (ring) of all polynomials in one variable \(t\) over \(F\text{.}\)

We begin with the following observation.

Observation A.1.1.

Let \(f(t)\in F[t]\) be a polynomial in one variable \(t\) of degree \(n\text{.}\) If \(x,y\in F\) then we have the Taylor’s formula:

\begin{equation} f(x+y)=f(x)+\frac{f^\prime(x)}{1}y+\frac{f^{\prime\prime}(x)}{2}y^2+\cdots+\frac{f^{(n)}(x)}{n!}y^n\tag{A.1.1} \end{equation}

Note that the above formula remains true if \(xy=yx\) in a ring with unity. In particular, if \(x,y\in M_n(F)\) such that \(xy=yx\) then the above formula holds. The following particular case will be of our interest.

\begin{equation} f(t)=f\left(a+(t-a)\right)=f(a)+\frac{f^\prime(a)}{1}(t-a)+\cdots+\frac{f^{(n)}(a)}{n!}(t-a)^n\tag{A.1.2} \end{equation}

Lemma A.1.2.

Let \(p(t)\in F[t]\) be such that \(p(t)=(t-a_1)(t-a_2)\cdots (t-a_n)\) for some \(a_i\in F\text{.}\) Then \(p(t)=t^n-s_1t^{n-1}+s_2t^{n-2}+\cdots+(-1)^ns_n\) for

\begin{equation*} s_k=\sum_{1\leq i_1<i_2<\cdots<i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}\quad\text{where}\; i_j\in\{1,2,\ldots,n\}. \end{equation*}

Remark A.1.3.

We call \(s_k\) the symmetric functions in \(a_{i_j}\text{.}\) Note that

\begin{align*} s_1\amp= a_1+a_2+\cdots+a_n\\ s_2\amp= a_1a_2+\cdots+a_{n-1}a_n\\ s_3\amp=a_1a_2a_3+\cdots+a_{n-2}a_{n-1}a_n\\ \amp\vdots\\ s_n\amp=a_1a_2\cdots a_n \end{align*}

Definition A.1.4. (Monic polynomial).

A polynomial \(p(t)=a_0+a_1t+\cdots+a_nt^n\in F[t]\) of degree \(n\) is said to be monic if \(a_n=1\) or in other words, the coefficient of the highest degree term of \(p(t)\) is \(1\text{.}\)

Definition A.1.5.

Let \(p(t)\in F[t]\) and \(a\in F\text{.}\) We say that \(a\) is a root of \(p(t)\) in \(F\) if \(p(a)=0\text{.}\)

Using (A.1.2) we get the following result.

Lemma A.1.6.

An element \(a\in F\) is a root of \(p(t)\in F[t]\) if and only if \(p(t)=(t-a)q(t)\) for some \(q(t)\in F[t]\text{.}\)

Definition A.1.7. (Split polynomial).

A non-constant polynomial \(p(t)\in F[t]\) is said to be split over \(F\) if \(p(t)\) is a product of polynomials of degree \(1\) in \(F[t]\text{.}\)

We define the multiplicity of root.

Definition A.1.8. (Multiplicity of a root).

If \(\alpha\in F\) is a root of a polynomial \(p(t)\in F[t]\) then, using Lemma A.1.6, we can write \(p(t)=(t-\alpha)^mq(t)\) where \(q(\alpha)\neq 0\text{.}\) We define \(m\) to be the multiplicity of a root \(\alpha\).

Using Lemma A.1.6 and induction on the degree of the polynomial we obtain the following result.

Lemma A.1.9.

A polynomial in one variable of degree \(n\) over a field \(F\) has at most \(n\) roots in \(F\text{.}\)

Theorem A.1.10. (Fundamental Theorem of Algebra).

A non-constant polynomial in \(\C[t]\) has a root in \(\C\) or, in the words, every non-constant polynomial in \(\C[t]\) splits over \(\C\text{.}\)

Linear Algebra: Companion notes

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