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Vieta’s Formula about the Sum of Roots of Polynomials

100%
Korniłowicz A., Pąk K.
Formalized Mathematics
|
2017
|
tom 25
|
nr 2
87-92
EN
In the article we formalized in the Mizar system [2] the Vieta formula about the sum of roots of a polynomial anxn + an−1xn−1 + ··· + a1x + a0 defined over an algebraically closed field. The formula says that [...] x1+x2+⋯+xn−1+xn=−an−1an $x_1 + x_2 + \cdots + x_{n - 1} + x_n = - {{a_{n - 1} } \over {a_n }}$ , where x1, x2,…, xn are (not necessarily distinct) roots of the polynomial [12]. In the article the sum is denoted by SumRoots.
2
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Positive characteristic analogs of closed polynomials

76%
Jędrzejewicz P.
Open Mathematics
|
2011
|
tom 9
|
nr 1
50-56
EN
The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.
3
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Characteristic of Rings. Prime Fields

76%
Schwarzweller C., Korniłowicz A.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
333-349
EN
The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.
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