Section A.1 Algebra of polynomials
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\label{Taylor-formula}\tag{A.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\label{Taylor-formula-particular}\tag{A.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.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.