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Ordered Rings and Fields

100%
Schwarzweller C.
Formalized Mathematics
|
2017
|
tom 25
|
nr 1
63-72
EN
We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].
2
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On Roots of Polynomials and Algebraically Closed Fields

100%
Schwarzweller C.
Formalized Mathematics
|
2017
|
tom 25
|
nr 3
185-195
EN
In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].
3
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The First Isomorphism Theorem and Other Properties of Rings

100%
Korniłowicz A., Schwarzweller C.
Formalized Mathematics
|
2014
|
tom 22
|
nr 4
291-301
EN
Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial
4
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Characteristic of Rings. Prime Fields

100%
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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