Khalid Javeed 25 Rings: Definition and Axioms
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Questions and Answers

What is the defining property of an integral domain?

  • Every element has a multiplicative inverse
  • Closure under addition and multiplication
  • Satisfies axioms A1 through A5 and M1 through M6 (correct)
  • Existence of multiplicative inverses
  • In a field F, what is the operation used to define division?

  • Multiplication
  • Inverse multiplication (correct)
  • Subtraction
  • Addition
  • Why are integers not considered a field?

  • Integers don't satisfy axioms A1 through A5
  • Integers don't have additive inverses
  • Not every integer has a multiplicative inverse (correct)
  • Integers are not closed under multiplication
  • What role do finite fields play in many cryptographic algorithms?

    <p>Finite fields are important for secure encryption and decryption</p> Signup and view all the answers

    What must be the order of a finite field (number of elements) according to the text?

    <p>A power of a prime number</p> Signup and view all the answers

    In a field, what does the existence of a multiplicative inverse mean?

    <p>Every non-zero element has a reciprocal</p> Signup and view all the answers

    In polynomial arithmetic with coefficients in a Galois Field (GF(p)), what is a fundamental property that allows for division?

    <p>Existence of multiplicative inverses</p> Signup and view all the answers

    What condition must the coefficient set satisfy for polynomial division to be possible?

    <p>Being an integral domain</p> Signup and view all the answers

    When performing polynomial arithmetic over a field, what is a key characteristic that makes division possible?

    <p>Existence of a multiplicative identity</p> Signup and view all the answers

    What property of a field ensures that polynomial division is consistent and well-defined?

    <p>Every non-zero element has an inverse</p> Signup and view all the answers

    Why is it crucial for a coefficient set to satisfy the properties of a field when performing polynomial division?

    <p>To ensure division is well-defined and consistent</p> Signup and view all the answers

    What set of numbers forms a field and guarantees that polynomial division will have consistent results?

    <p>GF(p) where p is a prime number</p> Signup and view all the answers

    In a ring R, which axiom states that R is an abelian group with respect to addition?

    <p>Axiom A1</p> Signup and view all the answers

    Which condition defines a commutative ring within the context of the text provided?

    <p>Commutativity of multiplication</p> Signup and view all the answers

    What characterizes an integral domain among rings according to the text?

    <p>Having a multiplicative identity</p> Signup and view all the answers

    If a, b belong to a ring R and ab = 0, what must be true based on the definition provided?

    <p>a = 0 or b = 0</p> Signup and view all the answers

    Which property ensures that for all a, b, c in a ring R, the following is valid: a(bc) = (ab)c?

    <p>Associativity of multiplication</p> Signup and view all the answers

    What effect does the absence of zero divisors have on the elements of an integral domain?

    <p>Each element must have a multiplicative inverse</p> Signup and view all the answers

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