Question

Show that the set \(\mathbb{Q}[i]\) of complex numbers of the form \(a+b i\), with \(a, b \in \mathbb{Q},\) is a subfield of \(\mathbb{C}\).

Short answer

Question: Show that the set of complex numbers of the form \(a + bi\), where \(a\) and \(b\) are rational numbers, denoted by \(\mathbb{Q}[i]\), is a subfield of the complex numbers \(\mathbb{C}\). Answer: To show that \(\mathbb{Q}[i]\) is a subfield of \(\mathbb{C}\), we demonstrated that it is closed under addition, subtraction, multiplication, and taking inverses (excluding the element 0), and that it contains 0 and 1. We proved that for any two elements in the set, their sum, difference, product, and inverses are also in the set. Since all the required conditions are satisfied, we can conclude that \(\mathbb{Q}[i]\) is a subfield of \(\mathbb{C}\).

Step-by-step solution

Closed under addition

Let \(x, y \in \mathbb{Q}[i]\). That means that \(x=a_1 + b_1i\) and \(y=a_2 + b_2i\), where \(a_1, a_2, b_1, b_2 \in \mathbb{Q}\). Now let's add these two complex numbers: \(x+y = (a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i\). Since the sum of two rational numbers is also a rational number, both \(a_1 + a_2\) and \(b_1 + b_2\) are in \(\mathbb{Q}\). Therefore, \(x+y \in \mathbb{Q}[i]\) and the set is closed under addition.

Closed under subtraction

Following a similar argument as in Step 1, let's subtract \(y\) from \(x\): \(x-y = (a_1 + b_1i) - (a_2 + b_2i) = (a_1 - a_2) + (b_1 - b_2)i\). Since the difference between two rational numbers is still a rational number, both \(a_1 - a_2\) and \(b_1 - b_2\) are in \(\mathbb{Q}\). Therefore, \(x-y \in \mathbb{Q}[i]\) and the set is closed under subtraction.

Closed under multiplication

Let's now multiply \(x\) and \(y\): \(xy = (a_1 + b_1i)(a_2 + b_2i) = a_1a_2 + a_1b_2i + a_2b_1i + b_1b_2i^2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i\). Since the product and sum of rational numbers are rational, both \((a_1a_2 - b_1b_2)\) and \((a_1b_2 + a_2b_1)\) are in \(\mathbb{Q}\). Therefore, \(xy \in \mathbb{Q}[i]\) and the set is closed under multiplication.

Closed under taking inverses

Let \(x = a + bi \in \mathbb{Q}[i]\). To find the inverse of \(x\), we can multiply the numerator and denominator of the fraction \(\frac{1}{x}\) by the conjugate of \(x\): \(x^{-1} = \frac{a - bi}{a^2 + b^2}\). Since the sum and product of rational numbers are rational, \(a - bi \in \mathbb{Q}[i]\) and \(\frac{1}{a^2 + b^2} \in \mathbb{Q}\) (assuming \(x \neq 0\)). Thus, \(x^{-1} \in \mathbb{Q}[i]\) and the set is closed under taking inverses.

Contains 0 and 1

Finally, we can observe that \(0 = 0 + 0i\) and \(1 = 1 + 0i\). As both 0 and 1 are rational, \(0, 1 \in \mathbb{Q}[i]\). Since all the required conditions are satisfied, we can conclude that the set \(\mathbb{Q}[i]\) is a subfield of \(\mathbb{C}\).

Key concepts

Closure Properties

The concept of closure properties is a foundational idea in mathematics that refers to the ability of a set to be self-contained under certain operations. Specifically, when discussing a subfield of the complex numbers, such as \(\mathbb{Q}[i]\), the closure properties ensure that when we perform operations like addition, subtraction, multiplication, and taking inverses, the result remains within the same set.

In the context of \(\mathbb{Q}[i]\), which is the set of complex numbers with rational components, we examined closure under various operations. The exercises demonstrated that the sum \(x+y\), the difference \(x-y\), and the product \(xy\) of any two elements \(x\) and \(y\) within \(\mathbb{Q}[i]\) also belonged to \(\mathbb{Q}[i]\), thus satisfying the closure properties. This is a crucial aspect when identifying a subset as a subfield – it must not 'escape' the set when operated upon.

Rational Numbers

Rational numbers form the backbone of the set \(\mathbb{Q}[i]\). They are numbers that can be expressed as the quotient of two integers, such as \(\frac{1}{2}\), \(\frac{3}{4}\), or \(\frac{-5}{3}\). These numbers are fundamental in closure properties of fields because they are closed under addition, subtraction, and multiplication.

As seen in our exercise, when we operate with complex numbers whose real and imaginary parts are both rational, the set retains its rational characteristics, making \(\mathbb{Q}[i]\) a suitable candidate for being a subfield of the complex numbers \(\mathbb{C}\). The simplicity and familiarity of rational numbers play an important role in this subfield, allowing for a straightforward understanding of complex arithmetic.

Complex Arithmetic

Complex arithmetic extends the traditional operations of addition, subtraction, and multiplication to accommodate the imaginary unit \(i\), which is defined by the property that \(i^2 = -1\). Within the subfield \(\mathbb{Q}[i]\), complex arithmetic allows us to combine and manipulate complex numbers while keeping their rational structure intact.

In the exercise, we performed addition, subtraction, and multiplication of complex numbers with rational components, and found that the results also had rational components, thus remaining in the set \(\mathbb{Q}[i]\). Such operations are foundational to the study of complex analysis and demonstrate how the algebraic rules we know from real numbers extend into the complex realm with elegance and order.

Field Theory

Field theory is a branch of algebra that studies the properties and structures of fields. A field is a set equipped with two operations, addition and multiplication, that satisfy certain axioms such as closure, associativity, identity and inverses for both operations, and distributivity of multiplication over addition.

Our set \(\mathbb{Q}[i]\) has been proven to be a subfield of \(\mathbb{C}\) through its closure properties, which is central to field theory. The existence of additive and multiplicative identities (0 and 1 respectively), the presence of additive and multiplicative inverses for every non-zero element, and the demonstrated closure under addition, subtraction, multiplication, and division by non-zero elements illustrate that \(\mathbb{Q}[i]\) obeys the axioms of a field. Hence, field theory provides the underlying framework that justifies the set \(\mathbb{Q}[i]\) as a well-defined mathematical structure within the realm of complex numbers.