Question

Explain why the set of integers modulo 10 under addition and multiplication is not a field.

   Explain why the set of integers modulo 10 under addition and multiplication is not a field.

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Darel W. Hardy, Fred… 2nd Edition

Chapter 9, Problem 9

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A field is a set equipped with two binary operations (typically called addition and multiplication) satisfying the following conditions: - It is an abelian group under addition. - It is a commutative ring with unity. - Every non-zero element has a Show more…

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Explain why the set of integers modulo 10 under addition and multiplication is not a field.

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Key Concepts

Field

A field is an algebraic structure in which addition, subtraction, multiplication, and division (except by zero) are well-defined and every nonzero element has a multiplicative inverse. This means that in a field, for every nonzero element, there exists another element that when multiplied together yields the multiplicative identity. The set of integers modulo 10 fails this requirement, as not every nonzero element has an inverse.

Zero Divisors

Zero divisors are elements in a ring such that when multiplied by some nonzero element, the result is zero. The existence of zero divisors indicates that there are nonzero elements that do not have multiplicative inverses, which prevents the structure from being a field. In the case of integers modulo 10, some numbers multiply to zero modulo 10, confirming the presence of zero divisors.

Modular Arithmetic

Modular arithmetic involves performing calculations with integers where numbers wrap around upon reaching a certain modulus. The structure of integers modulo n is widely studied in number theory and algebra. However, when the modulus is a composite number, such as 10, the resulting set may contain elements that behave as zero divisors, leading to failure of the multiplicative inverse property required for fields.

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