Algebraic Numbers

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This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.

[1] Michael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969.

[2] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.

[6] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[7] Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 2nd edition, 1989. Cambridge Studies in Advanced Mathematics.

[8] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.

[9] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.

[10] Masayoshi Nagata. Theory of Commutative Fields, volume 125. American Mathematical Society, 1985. Translations of Mathematical Monographs.

[11] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.

[12] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

[13] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[14] Oscar Zariski and Pierre Samuel. Commutative Algebra I. Springer, 2nd edition, 1975.

Formalized Mathematics

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researchers in the fields of formal methods and computer-checked mathematics


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