January 5, 2010 CHAPTER ONE ROOTS OF POLYNOMIALS §1
... of linear factors. Call a polynomial irreducible if and only if its degree is positive and it cannot be written as a product of more than one polynomial of positive degree; irreducible polynomials are the analogues of primes for integers. Thus, the set of polynomials irreducible over C coincides wit ...
... of linear factors. Call a polynomial irreducible if and only if its degree is positive and it cannot be written as a product of more than one polynomial of positive degree; irreducible polynomials are the analogues of primes for integers. Thus, the set of polynomials irreducible over C coincides wit ...
3 Complex Numbers
... This quadratic formula works regardless of the sign of the discriminant p2 − 4q, provided that we allow the roots to be complex, and take in account multiplicity. Namely, if p2 − 4q = 0, z 2 + pz + q = (z + p/2)2 and therefore the single root z = −p/2 has multiplicity two. If p2 − 4qp < 0 the roots ...
... This quadratic formula works regardless of the sign of the discriminant p2 − 4q, provided that we allow the roots to be complex, and take in account multiplicity. Namely, if p2 − 4q = 0, z 2 + pz + q = (z + p/2)2 and therefore the single root z = −p/2 has multiplicity two. If p2 − 4qp < 0 the roots ...
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.