📂Number Theory

Gaussian Integers

Definition ¹

$\mathbb{Z} [i] := \left\{ a + i b : a, b \in \mathbb{Z} \right\}$ is called a Gaussian Ring, and its elements are called Gaussian Integers.

Theorems

• [1]: $\overline{i} = i^{3}$
• [2]: $( a \pm ib ) + ( c \pm id) = (a \pm c) + i (b \pm d)$
• [3]: $( a + ib )( c + id) = (ac - bd) + i (ad + bc)$

Explanation

$i$ is a complex root of the quadratic equation $x^2 +1 = 0$, and $\mathbb{Z} [i]$ is an extension of the integer ring $\mathbb{Z}$. It is similar to the extension of the real number field $\mathbb{R}$ to the complex number field $\mathbb{C} = \mathbb{R} [i]$, and the principle behind it is not much different. There is no reason why complex numbers should be considered unthinkable, even when discussing integers where even irrational numbers are not taboo. In fact, they are simpler than irrational numbers.

Just as there are prime numbers among integers, there are Gaussian primes among Gaussian integers. On $\mathbb{Z} [i]$, the following conventional formulaic development is possible: $$\begin{align*} (7 + i2)(4 -i 2) =& (28 + 4) + i (- 14 +8 ) \\ =& 32 - i 6 \end{align*}$$ Also, given a natural number $n \in \mathbb{N}$, for a finite ring $\mathbb{Z}_{n}$, $\mathbb{Z}_{n}[i]$ can also be considered. For example, when $n = 7$, the development changes as follows: $$\begin{align*} (7 + i2)(4 -i 2) =& (28 + 4) + i (- 14 + 8 ) \\ =& 32 - i 6 \\ & \equiv 4 - i 6 \pmod{7} \\ & \equiv 4 + i \pmod{7} \end{align*}$$ Note how naturally the use of congruences comes into play. The desire to generalize $\mathbb{Z}$ to $\mathbb{Z} [i]$ is something natural to mathematicians, to an extent that it might be hard to explain in words. While it may be uncertain if it could compare to the innovations brought about by allowing $i$ in calculus, it is clear that number theory too has been enriched and made more beautiful. Just consider the Fundamental Theorem of Algebra, in $\mathbb{Z} [i]$, there is no messy talk of a $d$th degree polynomial equation having possibly more than $d$ solutions. With the introduction of complex numbers, it can simply be said to have exactly $d$ solutions.

A step further in the integer systems includes Eisenstein Integers.

The zero divisor graph of the Gaussian Ring has been studied by Osba.

Proofs

[1]

According to the definition of $i$ and the properties of conjugate, $$\overline{i} = -i = i^{3}$$

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[2]

Since $\mathbb{Z} [i]$ is a ring, and the additive operation satisfies the associative and commutative laws, $$\begin{align*} ( a \pm ib ) + ( c \pm id) =& a \pm ib + c \pm id \\ =& a \pm c + ib \pm id \\ =& (a \pm c) + i (b \pm d) \end{align*}$$

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[3]

From [2], $$\begin{align*} ( a + ib )( c + id) =& ac + ibc + iad + (- 1)bd \\ =& (ac - bd) + i (ad + bc) \end{align*}$$

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