Is Z5 a field?
In mathematics, a field is a set of elements that can be added, subtracted, multiplied, and divided (except for division by zero). It is a fundamental algebraic structure that is widely used in various branches of mathematics, including abstract algebra, number theory, and algebraic geometry. In this article, we will explore whether Z5, the set of integers modulo 5, forms a field.
To determine whether Z5 is a field, we need to check if it satisfies the field axioms. The field axioms include closure under addition and multiplication, the existence of additive and multiplicative identities, the existence of additive and multiplicative inverses for all non-zero elements, and the associativity and commutativity of addition and multiplication.
Firstly, let’s consider closure under addition and multiplication. The set Z5 consists of the elements {0, 1, 2, 3, 4}. When we add or multiply any two elements from this set, the result is also an element of Z5. For example, 2 + 3 = 5, which is congruent to 0 modulo 5, so 0 is also an element of Z5. This holds true for all combinations of elements in Z5, thus satisfying the closure property.
Next, we need to verify the existence of additive and multiplicative identities. In Z5, the additive identity is 0, as adding 0 to any element in the set leaves the element unchanged. Similarly, the multiplicative identity is 1, as multiplying any element by 1 leaves the element unchanged.
Now, let’s check the existence of additive and multiplicative inverses for all non-zero elements. In Z5, the additive inverse of an element a is the element b such that a + b = 0. For example, the additive inverse of 2 is 3, as 2 + 3 = 5, which is congruent to 0 modulo 5. The multiplicative inverse of an element a is the element b such that a b = 1. In Z5, the multiplicative inverse of 2 is 3, as 2 3 = 6, which is congruent to 1 modulo 5.
Finally, we need to ensure that the operations of addition and multiplication are associative and commutative. This property is satisfied in Z5, as it is in the set of integers.
In conclusion, Z5 satisfies all the field axioms, making it a field. Therefore, the answer to the question “Is Z5 a field?” is yes.
