Fields in Discrete Mathematics

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Groups, rings, and fields are important algebraic structures. In this article, we will explore the concept of fields in a simple way, go through its properties, and also provide some examples for a better understanding.

Fields are often discussed in Ring Theory, but here, we will see primarily what fields are, how they work, and what sets them apart from other algebraic structures.

What are Fields in Group Theory?

A field is an algebraic structure that consists of a non-empty set along with two binary operations. These are commonly known as addition and multiplication. For any set to be considered a field, it must satisfy several properties regarding these operations.

The three main requirements for a set to be a field are −

• Closure under addition and multiplication − If we add or multiply any two elements of the field, we should get another element of the field.
• Associativity of both operations − When adding or multiplying, the grouping of elements (brackets) does not matter. So, for any three elements, a, b, and c, we have (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) = a × (b × c).
• Commutativity of both operations − The order of elements that does not matter in either addition or multiplication. This means that a + b = b + a and a × b = b × a for all elements a and b in the field.

But besides these set of rules. There are a few more properties that define fields, which we will look into next.

Properties of a Field

For any set with binary operations to be a field, it must meet the following conditions −

• Existence of an Additive Identity − There must be an element which is denoted as 0 generally. When added to any element in the set, it leaves the element unchanged. In simple terms, a + 0 = a for all a in the set.
• Existence of a Multiplicative Identity − Similarly, there must be an element 1, such that multiplying it by any element does not change that element. That is, a × 1 = a for all a.
• Additive Inverses − For each element a in the field, there exists an inverse, denoted −a, such that a + (−a) = 0.
• Multiplicative Inverses for Non-Zero Elements − For every non-zero element a, there exists an element, often written as $\mathrm{\frac{1}{a}}$, such that $\mathrm{a \:\times\: \frac{1}{a} \:=\: 1}$.
• Distributive Property − The multiplication must distribute over addition, meaning that for all elements a, b, and c, we have a × (b + c) = (a × b) + (a × c).

If a set satisfies these properties, then it qualifies as a field. Let us see how these rules play out with real-world examples.

Common Examples of Fields

In this section, we have presented a set of common examples of Fields for a better understanding of the concept.

Real Numbers R

A very common example of a field is the set of real numbers. The real numbers, under the standard addition and multiplication must satisfy all the properties of a field. because:

• Closure − Adding or multiplying any two real numbers results in another real number.
• Commutativity and Associativity − Both addition and multiplication of real numbers are commutative and associative.
• Identities and Inverses − The additive identity is 0, and the multiplicative identity is 1. For any non-zero real number, there exists a multiplicative inverse (like 1/a).
• Distributive Law − The distributive property holds for real numbers, as a (b + c) = ab + ac.

Since all the field properties hold, the real numbers are considered a field.

Complex Numbers C

The set of complex numbers is another example of a field. This is also similar to the real numbers, the complex numbers satisfy all the necessary conditions −

• Closure − The sum or product of any two complex numbers is still a complex number.
• Commutativity and Associativity − These properties hold just like with real numbers.
• Identities and Inverses − The additive identity is 0 + 0i, and the multiplicative identity is 1 + 0i. Every non-zero complex number has a multiplicative inverse.
• Distributive Law − The distributive law applies to complex numbers too.

Thus, the complex numbers form a field.

Rational Numbers Q

The set of rational numbers, which includes all numbers that can be written as fractions $\mathrm{\frac{p}{q}}$ where p and q are integers, is also a field. Rational numbers satisfy the same field properties that real and complex numbers do −

• Closure − Any sum or product of two rational numbers is also a rational number.
• Identities and Inverses − Rational numbers have both additive and multiplicative identities and inverses.
• Distributive Property − The distributive property holds for rational numbers as well.

A Set That Is Not a Field: Integers Z

Consider the set of integers Z. While the integers satisfy some of the properties of a field, they fail to meet all the conditions.

• Additive Identity and Inverses − The integers have an additive identity (0), and every integer has an additive inverse (for example, the inverse of 5 is -5).
• Multiplicative Identity − There is a multiplicative identity (1), but the problem lies in inverses.
• Multiplicative Inverses − Not every non-zero integer has a multiplicative inverse. For example, 2 does not have an integer x inverse because there is no integer such that $\mathrm{2 \:\times\: x \:=\: 1}$.

Since not all integers have multiplicative inverses, Z is not a field.

Use Cases of Fields

Fields are such powerful tool in mathematics. Fields have many applications in discrete mathematics and computer science. For example, fields are used in cryptography to ensure secure communication. The widely-used RSA encryption relies heavily on the properties of fields and modular arithmetic, a concept closely related to field theory.

In coding theory, fields are used in error-detecting and error-correcting codes, which help in transmitting data accurately over unreliable channels. We can find applications in linear algebra, where field properties help in solving systems of linear equations.

Conclusion

In this chapter, we elaborated the concept of fields in discrete mathematics. We started by understanding what a field is, the properties that define it, and then looked at some examples like real numbers, complex numbers, and rational numbers.

We also provided an example of a set, the integers, that do not form a field. Using the examples, we highlighted how fields behave and why they are significant in various areas of mathematics. We also touched upon how fields are applied in areas like cryptography and coding theory.