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Some Algebraic Properties of Polynomial Rings

100%

Schwarzweller C., Korniłowicz A.

Formalized Mathematics

2016

tom 24

nr 3

227-237

In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.

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On rational radii coin representations of the wheel graph

88%

Agnarsson G., Dunham J.

Discussiones Mathematicae - General Algebra and Applications

2013

tom 33

nr 2

167-199

A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the cosinus of the internal angles formed by the center coin and two consecutive petals of the flower. In that process we also derive a recursion that these irreducible polynomials satisfy.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Formalized Mathematics

1 Agnarsson G.

1 Dunham J.

1 Korniłowicz A.

1 Schwarzweller C.

1 2016

1 2013

Wyniki wyszukiwania

Some Algebraic Properties of Polynomial Rings

On rational radii coin representations of the wheel graph