What Do You Know About The Root Datum?

An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations. The multiplication and inversion operations on an algebraic group are given by regular maps on the variety, which means they can be expressed as polynomial functions. This makes the algebraic group a natural and well-behaved object in algebraic geometry. Compact group, Lie group, and planar group do not necessarily have the property of being defined by polynomial equations, so they do not fit the description given in the question.

The special linear group is an example of a reductive group. A reductive group is a type of algebraic group that satisfies certain properties related to its representation theory. The special linear group is a subgroup of the general linear group consisting of all invertible matrices with determinant equal to 1. It is a reductive group because it can be decomposed into a direct product of a torus and a semisimple group, which is a key property of reductive groups.

A univariate polynomial is absolutely irreducible if and only if its degree is one. This means that the polynomial cannot be factored into polynomials of lower degree with coefficients in the same field. A polynomial of degree zero is a constant, and a constant polynomial is trivially irreducible. However, a polynomial of degree one is a linear polynomial, which cannot be factored into lower degree polynomials. Therefore, a univariate polynomial is absolutely irreducible if and only if its degree is one.

The jet group is a generalization of the general linear group that applies to Taylor polynomials instead of vectors at a point. It is a mathematical structure that captures the behavior of functions near a point by considering their Taylor expansions. The jet group allows for the study of higher-order derivatives and provides a framework for understanding the local behavior of functions in a more detailed way.

A root system is a configuration of vectors in an Euclidean space that satisfies certain geometrical properties. It is a set of vectors in a vector space that spans the space and has specific symmetry properties. Root systems are used in various branches of mathematics, such as Lie theory and algebraic geometry, to study the structure of groups and algebras. They play a crucial role in understanding the symmetries of geometric objects and have applications in physics, computer science, and other fields.

A Borel subgroup is a maximal Zariski closed and connected solvable algebraic subgroup. It is a subgroup that is closed under Zariski topology, connected, and solvable, meaning that it can be generated by a sequence of subgroups, each of which is normal in the previous subgroup and has abelian quotient. Borel subgroups play an important role in algebraic geometry and Lie theory, and they have many applications in various areas of mathematics.

An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. This means that it cannot be broken down any further and has no factors other than itself and the constant term. Therefore, an irreducible polynomial is the correct answer to the question.

An elliptic curve is a plane algebraic curve defined by an equation that is non-singular. This means that the curve does not have any singular points, where the derivative of the equation is zero. Elliptic curves have a rich mathematical structure and are widely used in cryptography and number theory.

The irreducibility of a polynomial is determined by the nature of its coefficients. The nature of the coefficients refers to whether they are rational or irrational numbers. If the coefficients are rational numbers, the polynomial may be reducible, meaning it can be factored into lower degree polynomials with rational coefficients. On the other hand, if the coefficients are irrational numbers, the polynomial is more likely to be irreducible, meaning it cannot be factored into lower degree polynomials with rational coefficients. Therefore, the nature of the coefficient is the determining factor for the irreducibility of a polynomial.

Complex numbers are an example of an algebraically closed field because they contain both real and imaginary numbers. In a complex number, the real part represents the horizontal axis, while the imaginary part represents the vertical axis. Complex numbers satisfy all algebraic equations, including polynomial equations, making them algebraically closed. Real numbers, rational numbers, and irrational numbers do not have this property, as they do not include both real and imaginary components.