Lecture

Polynomials: Theory and Operations

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Description

This lecture covers the theory and operations related to polynomials, including concepts such as ideals, minimal polynomials, irreducibility, and the Euclidean algorithm. It explores properties of polynomials, factorization, valuations, and the Bezout theorem.

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Related concepts (32)

Galois theory

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials.

January

January is the first month of the year in the Julian and Gregorian calendars and is also the first of seven months to have a length of 31 days. The first day of the month is known as New Year's Day. It is, on average, the coldest month of the year within most of the Northern Hemisphere (where it is the second month of winter) and the warmest month of the year within most of the Southern Hemisphere (where it is the second month of summer). In the Southern hemisphere, January is the seasonal equivalent of July in the Northern hemisphere and vice versa.

Factorization of polynomials

In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension.

Factorial

In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah.

Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly.

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