Is Z3 a Field?
In the realm of mathematics, the concept of a field is fundamental to understanding various algebraic structures. A field is a set equipped with two operations, addition and multiplication, that satisfy certain axioms. These axioms include commutativity, associativity, the existence of an additive and a multiplicative identity, and the existence of additive and multiplicative inverses for all elements except the additive identity. The question “Is Z3 a field?” arises from the desire to understand whether the set of integers modulo 3, denoted as Z3, meets the criteria of a field.
Z3 is the set of integers {0, 1, 2}, where the operations of addition and multiplication are performed modulo 3. In other words, when adding or multiplying two elements of Z3, the result is the remainder obtained after dividing the sum or product by 3. For example, 1 + 2 = 3, which is equivalent to 0 in Z3, since 3 divided by 3 has a remainder of 0.
To determine whether Z3 is a field, we must examine the axioms of a field and verify if they are satisfied by the operations in Z3. The first axiom is commutativity, which states that the order of the operands does not affect the result of an operation. In Z3, addition and multiplication are both commutative, as can be seen from the following examples:
1 + 2 = 2 + 1 = 0 (mod 3)
2 1 = 1 2 = 2 (mod 3)
The second axiom is associativity, which asserts that the grouping of operands does not affect the result of an operation. Addition and multiplication in Z3 are also associative, as demonstrated by the following:
(1 + 2) + 3 = 1 + (2 + 3) = 0 + 0 = 0 (mod 3)
(2 1) 2 = 2 (1 2) = 2 2 = 1 (mod 3)
The third and fourth axioms require the existence of an additive and a multiplicative identity, as well as additive and multiplicative inverses for all elements except the additive identity. In Z3, the additive identity is 0, and the multiplicative identity is 1. However, the existence of multiplicative inverses for all elements except 0 is where Z3 fails to meet the criteria of a field.
Consider the element 2 in Z3. To find its multiplicative inverse, we need an element x such that 2 x = 1 (mod 3). However, no such element exists in Z3. The product of 2 and 1 is 2, and the product of 2 and 2 is 1 (mod 3), but 1 is not the multiplicative inverse of 2, as 2 1 ≠ 1 (mod 3). Similarly, 2 does not have a multiplicative inverse in Z3.
Since Z3 does not satisfy the requirement of having multiplicative inverses for all elements except 0, it is not a field. This conclusion highlights the importance of carefully examining the axioms of a field when determining whether a given set equipped with operations is indeed a field.
