What Do You Know About Absolutely Irreducible Polynomials?

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What Do You Know About Absolutely Irreducible Polynomials? - Quiz

For one, the title is self explanatory; this quiz assess your knowledge of absolutely irreducible polynomials.
In the field of mathematics, these are polynomials that are irreducible or indivisible over a complex numbe, the complex field, or every algebraic extension of their respective selves.
In general, they are multivariate polynomials defined over the rational numbers.


Questions and Answers
  • 1. 

    For a rational number to be absolutely irreducible, what field must it pass through? 

    • A.

      Simple field

    • B.

      Complex field

    • C.

      Compound field

    • D.

      Multiple field

    Correct Answer
    B. Complex field
    Explanation
    A rational number is said to be absolutely irreducible if it cannot be further reduced or simplified. In order for a rational number to be absolutely irreducible, it must pass through the complex field. The complex field consists of all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. This field allows for the representation of all rational numbers, as well as irrational numbers and complex numbers. Therefore, passing through the complex field ensures that a rational number cannot be further simplified or reduced, making it absolutely irreducible.

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  • 2. 

    For field K, how do we describe if the field is irreducible? 

    • A.

      Algebraic extension of K

    • B.

      Geometric extension of K

    • C.

      Trigonometry rate of K

    • D.

      Multiple patterns of K

    Correct Answer
    A. Algebraic extension of K
    Explanation
    An algebraic extension of a field K is a field L that contains K and every element of L is a root of a polynomial with coefficients in K. This means that every element in L can be obtained by performing algebraic operations (addition, subtraction, multiplication, and division) on elements in K. Therefore, if a field K is described as an algebraic extension, it means that all elements in K can be expressed as roots of polynomials with coefficients in K, indicating that K is irreducible.

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  • 3. 

    What is the synonym of an absolutely irreducible algebraic set? 

    • A.

      Algebraic variety

    • B.

      Geometric variety

    • C.

      Algebraic expression

    • D.

      Geometric expression

    Correct Answer
    A. Algebraic variety
    Explanation
    An absolutely irreducible algebraic set is a set that cannot be expressed as the union of two proper algebraic subsets. An algebraic variety is a geometric object that is defined as the zero set of a collection of polynomial equations. Therefore, an algebraic variety is a synonym for an absolutely irreducible algebraic set.

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  • 4. 

    Which algebraic group is the absolutely irreducible applied to? 

    • A.

      Geometric representation

    • B.

      Algebraic representation

    • C.

      Linear representation

    • D.

      Random representation

    Correct Answer
    C. Linear representation
    Explanation
    The correct answer is Linear representation. In mathematics, an algebraic group is said to be absolutely irreducible if it cannot be decomposed into a direct product of nontrivial algebraic subgroups. The linear representation of an algebraic group refers to the group action on a vector space, where the group elements are represented by linear transformations. This representation is often used to study the structure and properties of the algebraic group.

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  • 5. 

    How do we represent the absolutely irreducible of an univariate polynomial of degrees greater or equal to 2?

    • A.

      In a linear form

    • B.

      In a geometric pattern

    • C.

      By polynomial

    • D.

      They have no absolutely irreducible value

    Correct Answer
    D. They have no absolutely irreducible value
    Explanation
    The correct answer is "They have no absolutely irreducible value." This means that univariate polynomials of degrees greater than or equal to 2 cannot be represented in a way that simplifies or reduces them further. They cannot be factored into simpler forms or expressed in terms of simpler functions. This is in contrast to linear polynomials, which can be represented in a simpler form. Similarly, geometric patterns and polynomial representations do not apply to these higher-degree polynomials.

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  • 6. 

    What do we call the decomposition of a multivariate polynomial as a product of absolutely irreducible polynomials?

    • A.

      Absolute factorization

    • B.

      Absolute decomposition

    • C.

      Absolute disintegration

    • D.

      Absolute computation

    Correct Answer
    A. Absolute factorization
    Explanation
    The decomposition of a multivariate polynomial as a product of absolutely irreducible polynomials is called absolute factorization.

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  • 7. 

    How do we represent the absolutely irreducible algorithms? 

    • A.

      Geometric progressions

    • B.

      Polynomials

    • C.

      Arithmetic progressions

    • D.

      Trigonometric rates

    Correct Answer
    B. Polynomials
    Explanation
    Polynomials can represent absolutely irreducible algorithms because they are mathematical expressions that involve variables and coefficients, and can be used to model and solve various problems. Polynomials can be manipulated using algebraic operations such as addition, subtraction, multiplication, and division, allowing for the creation of complex algorithms. Additionally, polynomials have the property of irreducibility, meaning they cannot be factored into simpler polynomials. This makes them suitable for representing algorithms that cannot be further simplified or broken down into smaller components.

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  • 8. 

    What are absolutely irreducible polynomials also called?

    • A.

      Non-constant polynomials

    • B.

      Constant polynomials

    • C.

      Random polynomials

    • D.

      Fractional polynomials

    Correct Answer
    A. Non-constant polynomials
    Explanation
    Absolutely irreducible polynomials are non-constant polynomials that cannot be factored into smaller polynomials over a given field. These polynomials are called non-constant because they are not constant polynomials, which are polynomials with a single term. They are also not random polynomials or fractional polynomials, as these terms do not accurately describe the nature of absolutely irreducible polynomials.

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  • 9. 

    What does the property of irreducibility depend on?

    • A.

      Number of coefficients

    • B.

      Nature of coefficients

    • C.

      Type of coefficients

    • D.

      Value of coefficients

    Correct Answer
    B. Nature of coefficients
    Explanation
    The property of irreducibility depends on the nature of coefficients. Irreducibility refers to the inability to factorize a polynomial into non-trivial factors. The nature of coefficients, such as whether they are rational or irrational, real or complex, can determine whether a polynomial is irreducible or not. For example, if the coefficients are rational, it is possible to factorize the polynomial using rational roots theorem. On the other hand, if the coefficients are irrational or complex, it may not be possible to factorize the polynomial further, making it irreducible.

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  • 10. 

    What sole condition describes a univariate polynomial that is absolutely irreducible? Its degree must be... 

    • A.

      4

    • B.

      3

    • C.

      2

    • D.

      1

    Correct Answer
    D. 1
    Explanation
    A univariate polynomial is absolutely irreducible if it cannot be factored into polynomials of lower degree with coefficients in the same field. In this case, the correct answer is 1, meaning that the polynomial must have a degree of 1. This means that the polynomial is a linear equation and cannot be factored further.

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  • Mar 19, 2023
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  • May 31, 2018
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    AdeKoju
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