What Do You Know About The Root Datum?
Considered and introduced by Michel Demazure in the 1970s, the root datum is a widely used approach in the group theory branch of mathematics.
It is a speculation of a root system or framework that decides the gathering up to isomorphism.
This generally means the the root datum of a linked break up reductive algebraic group on the field.
- 1.
Which is an example of a reductive group?
- A.
Projective group
- B.
Special linear group
- C.
Abelian group
- D.
Compact group
Correct Answer
B. Special linear groupExplanation
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.Rate this question:
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- 2.
What is a configuration of vectors in an Euclidean space that satisfies certain geometrical properties?
- A.
Root system
- B.
Datum system
- C.
Plane system
- D.
Curve system
Correct Answer
A. Root systemExplanation
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.Rate this question:
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- 3.
What is a maximal Zariski closed and connected solvable algebraic subgroup?
- A.
Compact subgroup
- B.
Projective subgroup
- C.
Planar subgroup
- D.
Borel subgroup
Correct Answer
D. Borel subgroupExplanation
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.Rate this question:
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- 4.
Which is an example of an algebraically closed field?
- A.
Complex numbers
- B.
Real numbers
- C.
Rational numbers
- D.
Irrational numbers
Correct Answer
A. Complex numbersExplanation
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.Rate this question:
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- 5.
What is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials?
- A.
Reducible polynomial
- B.
Irreducible polynomial
- C.
Constant polynomial
- D.
Projective polynomial
Correct Answer
B. Irreducible polynomialExplanation
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.Rate this question:
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- 6.
Which is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety?
- A.
Compact group
- B.
Lie group
- C.
Algebraic group
- D.
Planar group
Correct Answer
C. Algebraic groupExplanation
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.Rate this question:
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- 7.
What is a plane algebraic curve defined by an equation that is non-singular?
- A.
Elliptic curve
- B.
Cubic curve
- C.
Quadric curve
- D.
Projective curve
Correct Answer
A. Elliptic curveExplanation
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.Rate this question:
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- 8.
What is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point?
- A.
Jet group
- B.
Lie group
- C.
Canort group
- D.
Nilpotent group
Correct Answer
A. Jet groupExplanation
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.Rate this question:
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- 9.
What determines the irreducibility of a polynomial?
- A.
Nature of coefficient
- B.
Shape of slope
- C.
Number of coefficient
- D.
Nature of quotient
Correct Answer
A. Nature of coefficientExplanation
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.Rate this question:
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- 10.
A univariate polynomial is absolutely irreducible if and only if its degree is what?
- A.
Zero
- B.
One
- C.
Two
- D.
Three
Correct Answer
B. OneExplanation
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.Rate this question:
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