We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) roots.

[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds ||z| | = |z|. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) · (x2 + y2i) = (x1 · x2 − y1 · y2) + (x1 · y2 + x2 · y1)i. (3) For every real number r holds (r + 0i) · i = 0 + ri. (4) For every real number r holds |r + 0i | = |r|. (5) For every element z of C such that |z | � = 0 holds |z | + 0i = z |z|+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x + yiCF yielding an element of CF is defined as follows: (Def. 1) x + yiCF = x + yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0 + 1i and iCF = 0 + 1iCF.

Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.

The scheme Regr without 0 concerns a unary predicateP, and states that: P[1] provided the parameters meet the following requirements: • There exists a non empty natural number k such thatP[k], and • For every non empty natural number k such that k � = 1 andP[k] there exists a non empty natural number n such that n < k andP[n]. The following propositions are true: (3) 1 For every element z of C holds ℜ(z) ≥ −|z|. (4) For every element z of C holds ℑ(z) ≥ −|z|. (5) For every element z of CF holds ℜ(z) ≥ −|z|. (6) For every element z of CF holds ℑ(z) ≥ −|z|. (7) For every element z of CF holds |z | 2 = ℜ(z) 2 + ℑ(z) 2. (8) For all real numbers x1, x2, y1, y2 such that x1 + x2iCF = y1 + y2iCF holds x1 = y1 and x2 = y2. (9) For every element z of CF holds z = ℜ(z)+ℑ(z)iCF. (10) 0CF = 0+0iCF. (12) 2 For every unital non empty groupoid L and for every element x of L holds power L (x, 1) = x. (13) For every unital non empty groupoid L and for every element x of L holds power L (x, 2) = x · x. (14) Let L be an add-associative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n> 0, then power L (0L, n) = 0L. 1 The propositions (1) and (2) have been removed. 2 The proposition (11) has been removed.

Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The construction of scalar product in quotient vector space from non negative hermitan functions is the main result of the article.

and q a = 1holds a = 0. (2) Forallnaturalnumbers a, k, randforeveryrealnumber xsuchthat 1 < xand 0 < kholds x a·k+r = x a · x a·(k− ′ 1)+r (3) Forallnaturalnumbers q, a, bsuchthat 0 < aand 1 < qand q a − ′ 1| q b − ′ 1holds a|b. (4) Forallnaturalnumbers n, qsuchthat 0 < qholds q n = q n. 1 ThisworkhasbeensupportedbyNSERCGrantOGP9207.