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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].
Christoph Schwarzweller. "On Roots of Polynomials and Algebraically Closed Fields." Formalized Mathematics 25.3 (2017): 185-195. <http://eudml.org/doc/288477>.
@article{ChristophSchwarzweller2017, abstract = {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].}, author = {Christoph Schwarzweller}, journal = {Formalized Mathematics}, keywords = {commutative algebra; polynomials; algebraic closed fields}, language = {eng}, number = {3}, pages = {185-195}, title = {On Roots of Polynomials and Algebraically Closed Fields}, url = {http://eudml.org/doc/288477}, volume = {25}, year = {2017}, }
TY - JOUR AU - Christoph Schwarzweller TI - On Roots of Polynomials and Algebraically Closed Fields JO - Formalized Mathematics PY - 2017 VL - 25 IS - 3 SP - 185 EP - 195 AB - 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]. LA - eng KW - commutative algebra; polynomials; algebraic closed fields UR - http://eudml.org/doc/288477 ER -