Linear algebra (Osnabrück 2024-2025)/Part I/Important theorems

???:Binomial theorem

Let ${}a,b$ be elements of a field and let ${}n$ denote a natural number. Then

$(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}$

holds.

???:Euclidean division (polynomial ring)

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P,T\in K[X]$ be polynomials with ${}T\neq 0$. Then there exist unique polynomials ${}Q,R\in K[X]$ such that

$P=TQ+R{\text{ and with }}\operatorname {deg} \,(R)<\operatorname {deg} \,(T){\text{ or }}R=0.$

???:Linear factor and zero of a polynomial

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P\in K[X]$ be a polynomial and ${}a\in K$. Then ${}a$ is a zero of ${}P$ if and only if ${}P$ is a multiple of the linear polynomial ${}X-a$.

???:Number of zeroes of a polynomial

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P\in K[X]$ be a polynomial (${}\neq 0$) of degree ${}d$. Then ${}P$ has at most ${}d$ zeroes.

???:Fundamental theorem of algebra

Every nonconstant polynomial ${}P\in \mathbb {C} [X]$ over the complex numbers has a zero.

???:Interpolation theorem for polynomials

Let ${}K$ be a field, and let ${}n$ different elements ${}a_{1},\ldots ,a_{n}\in K$ and ${}n$ elements ${}b_{1},\ldots ,b_{n}\in K$ are given. Then there exist a unique polynomial ${}P\in K[X]$ of degree ${}\leq n-1$, such that ${}P{\left(a_{i}\right)}=b_{i}$ holds for all ${}i$.