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Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure. In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
Parsons problems are a form of an objective assessment in which respondents are asked to choose from a selection of code fragments, some subset of which comprise the problem solution. The Parsons problem format is used in the learning and teaching of computer programming. Dale Parsons and Patricia Haden of Otago Polytechnic developed Parsons's ...
If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x 2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root.
In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of ...
To see this, choose a monic irreducible polynomial f(X 1, ..., X n, Y) whose root generates N over E. If f(a 1, ..., a n, Y) is irreducible for some a i, then a root of it will generate the asserted N 0.) Construction of elliptic curves with large rank. [2] Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's ...
It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters (they are sometimes reducible for special parameter values; see below), but this was finally proved by Nishioka (1988) and Hiroshi Umemura . These six second order nonlinear differential equations are ...