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

???:Binomial theorem

Let be elements of a field and let denote a natural number. Then

holds.


???:Euclidean division (polynomial ring)

Let be a field and let be the polynomial ring over . Let be polynomials with . Then there exist unique polynomials such that


???:Linear factor and zero of a polynomial

Let be a field and let be the polynomial ring over . Let be a polynomial and . Then is a zero of if and only if is a multiple of the linear polynomial .


???:Number of zeroes of a polynomial

Let be a field and let be the polynomial ring over . Let be a polynomial () of degree . Then has at most zeroes.


???:Fundamental theorem of algebra

Every nonconstant polynomial over the complex numbers has a zero.


???:Interpolation theorem for polynomials

Let be a field, and let different elements and elements are given. Then there exist a unique polynomial of degree , such that holds for all .