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I have below questions concerning Gaussian integers and are as follows. In each question, I am presenting my understanding for vetting, or asking a direct question:

Gaussian Integers are formed with two coordinates in the complex plane, x (first) being real and the y (second) coordinate being imaginary. They can be represented as a lattice (grid) of coordinates.

i) For a Gaussian integer, the lattice points are formed by considering all points in the grid formed by x and y coordinates possible as the multiples thereof. So, if $2+3i$ is a Gaussian integer, then lattice is formed by ($a+bi$)($2+3i$), for $a,b \in Z$.

Also, by using the property of the distributive law being followed in Gaussian integers (plane), can add up any 'x'(real) and 'y'(imaginary) multiple (or, any linear combination thereof) of Gaussian integers.

Say, for $2+3i$ as Gaussian integer, let the multiples be formed by $3$, and imaginary point $2i$. Then, $3*(2+3i)$ is a Gaussian integer, and also $2i*(2+3i)$, and so is $(3+2i)*(2 + 3i)$.

Let us consider the number: $5-3i$, and it being a multiple of $2+3i$ means if it lies on the lattice of $2+3i$.

Geometrically, the question of whether $5-3i$ is a multiple of $2+3i$ is if it lies on the lattice formed by $2+3i$. Algebraically, it means being a linear combination is same as being a multiple.

ii) What Gaussian integers are multiples of $i$?

Multiplication of any linear combination of real and imaginary integers (i.e, $1, i$) with $i$ are multiple of $i$. It seems a vague question, and seemingly must have an algebraic answer as: $(a+ib).i$, where $a, b \in Z$. The reason being that this should give all possible linear combinations.

(iii) How to find a quadratic equation being satisfied by $i - 5$, and the geometrical significance of the same.

For a polynomial equation, the irrational roots occur in conjugate pairs, so the other root is : $i +5$.

For the given Gaussian integer, the sum of roots of $i-5$ & $-i-5$ is = -10; while the product of roots is: $26$.

So, the quadratic equation is : $X^2 -10X +26 =0$.

But, the geometrical significance of such quadratic equation being satisfied by a Gaussian integer is not clear.

jiten
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  • For part (ii), all Gaussian integers are multiples of $i$. In fact $i$ is invertible (also known as a unit in the ring of Gaussian integers). Hence for any element $a + bi$, one can find $a + bi = i(c + di)$, by multiplying the equation by $-i$, the inverse of $i$. You get $c + di = -i(a + bi) = b - ai$. – Tob Ernack Nov 16 '17 at 15:57
  • I would have loved a proof. – jiten Nov 16 '17 at 16:03
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    You can verify by checking that $i(b - ai) = a + bi$. – Tob Ernack Nov 16 '17 at 16:34
  • @TobErnack But, still I feel more could be done. May be, it is over. – jiten Nov 16 '17 at 22:31

1 Answers1

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$(i)$ To check if $5-3i$ is a multiple of $2+3i$.

Let $$5-3i=(a+bi)(2+3i)$$

where $a,b \in \mathbb{Z}$.

$$2a-3b=5\tag{1}$$ $$3a+2b=-3\tag{2}$$

Multiplying $(1)$ by $3$, $$6a-9b=15\tag{3}$$

Multiplyng $(2)$ by $2$, $$6a+4b=-6\tag{4}$$

SUbtracting $(4)$ from $(3)$, we have

$$-13b=21$$

but no such $b$ can exist. Hence $5-3i$ is not a multiple of $2+3i$.

$(ii)$ Since $$a+bi=(b-ai)i$$

The multiples of $i$ are exactly the set of all Gaussian integers.

$(iii)$ We need more information, in particular, must the coefficient be real. The conjugate pair argument only works if the coefficients are real. Otherwise, if you don't mind your integer being any Gaussian integer, the quadratic equation can take the form of $$(x-i+5)(x-a-bi)=0$$

Siong Thye Goh
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