Are the Rationals a Field?
The question of whether the set of rational numbers, denoted by Q, forms a field is a fundamental topic in abstract algebra. A field is a mathematical structure that consists of a set of elements together with two binary operations, addition and multiplication, which satisfy certain axioms. These axioms include the existence of an additive identity, a multiplicative identity, commutativity, associativity, distributivity, and the existence of additive inverses and multiplicative inverses for all non-zero elements. In this article, we will explore whether the rationals satisfy these axioms and, consequently, whether they form a field.
Firstly, let’s consider the existence of an additive identity, which is the number zero. In the set of rational numbers, zero is indeed an element, and it satisfies the additive identity property, as adding zero to any rational number results in the same number. This is evident from the definition of rational numbers, which are fractions of the form a/b, where a and b are integers and b is not equal to zero. Adding zero to a rational number a/b simply results in a/b + 0 = a/b.
Next, we need to establish the existence of a multiplicative identity, which is the number one. Similarly, the rational number 1 is an element of Q, and it satisfies the multiplicative identity property, as multiplying any rational number by one results in the same number. For any rational number a/b, multiplying it by one yields a/b 1 = a/b.
Now, let’s consider the commutative and associative properties of addition and multiplication. The commutative property states that changing the order of the operands in an addition or multiplication operation does not change the result. In the case of rational numbers, both addition and multiplication are commutative. For example, for any two rational numbers a/b and c/d, we have a/b + c/d = c/d + a/b and a/b c/d = c/d a/b. The associative property states that the grouping of operands in an addition or multiplication operation does not affect the result. Again, both addition and multiplication of rational numbers are associative, as demonstrated by the following examples: (a/b + c/d) + e/f = a/b + (c/d + e/f) and (a/b c/d) e/f = a/b (c/d e/f).
The distributive property states that multiplication distributes over addition. For rational numbers, this property holds true as well. For any three rational numbers a/b, c/d, and e/f, we have a/b (c/d + e/f) = (a/b c/d) + (a/b e/f).
Finally, we need to consider the existence of additive inverses and multiplicative inverses for all non-zero elements in the set of rational numbers. An additive inverse of a rational number a/b is a rational number -a/b, which, when added to a/b, results in zero. Similarly, a multiplicative inverse of a rational number a/b is a rational number b/a, which, when multiplied by a/b, results in one. It is easy to verify that the additive inverses and multiplicative inverses exist for all non-zero rational numbers.
In conclusion, the set of rational numbers satisfies all the axioms of a field. Therefore, we can confidently say that the rationals are indeed a field. This property makes the rational numbers a useful and fundamental mathematical structure in various branches of mathematics, including algebra, number theory, and analysis.
