Now, the fun begins. What happens when the complex numbers investigated above are taken to successive powers? Let us take the most simple example: Modulus 5.

1

2

3

4

5

1

1

1

1

1

1

2

2

4

3

1

2

3

3

4

2

1

3

4

4

1

4

1

4

Intheleftmostcolumnstandseachresidueof

modulus5,whilethetoprowsignifies

thepowertowhicheachvalueoftheformer

israised.

Remember, the leftmost column has been previously mapped onto the complex plane, corresponding to 1, i, -i, -1 respectively, or 1, -i, i, -1 respectively. The properties of the column labeled "2" were used earlier, namely 2^{2} º 4 and 3^{2} º 4. Now, how does one go about representing these as well as the properties of the higher powers, so that characteristics of such transformations can be investigated geometrically?

Here is a representation of each residue of modulus 5, squared:

From left to right is first a representation of i^{2}, with the result, -1, pointed to by an arrow. Next is -1^{2}, then -i^{2}, and 1^{2} º 1, an action indicated by a circular arrow. Next is a combination of all four. Finally, the rightmost is a figure resembling a very square pox infected patient. The large pock is an indication of a residue of the fourth power, or a biquadratic residue: in this case, i^{4} º -1^{4} º -i^{4} º 1^{4}. The medium-sized pock represents a squared number, -1^{2}, or a quadratic residue which is not biquadratic. The smallest pox represent numbers which are quadratic non-residues (i, -i) of which will be elaborated later. This is a type of representation which will occur several times throughout this writing, to geometrically show the biquadratic characteristics which each modulus has, collectively known as the modulus' biquadratic character.

Notice that by squaring every residue once, the biquadratic character of modulus 5 is also immediately had; namely, the biquadratic residues, are those residues which are squares of a squared number. In this case, 1 º -1^{2}, where -1 º i^{2}. In the picture above, 1 is being pointed at by an arrow from -1, which is itself begin pointed at by an arrow originating from both i and -i.

Also of note, is that the number 1 here is a residue of modulus p=5, as well as a quadratic, cubic and biquadratic residue of p=5. Though it is the same number, can the outcome of higher degrees of change be the same? In other words, are 1, 1^{2}, 1^{3} and 1^{4} really the same number?

Now look at modulus p=13 Shown on the left, indicated by the arrows, is the action of each residue of modulus 13 squared. The characteristics of the four unities, 1, -1, i and -i are not shown here, because they are the same as in modulus 5, and will be for several other moduli, i.e. 1 º i^{4}, -1 º i^{2}.

The result of squaring can be seen more clearly if the residues are separated into three sets of associates, indicated in the above diagram by the three different colors.

For example, the purple set consists of the numbers 1-i, -1-i, -1+i, and 1+i. Each set has a biquadratic character which is identical with that of the set of unities, 1, -1, i, -i, which constitute the residues of modulus 5. Namely, each set contains one biquadratic residue, one quadratic residue, and two quadratic non-residues.

Again, a similar event occurs in the case of p=29.

You can see, that the characteristic of action, represented by the different sized dots, define sets of numbers which are associated with each other, which, not coincidently, are the numbers of equal distance from 0, called associates.

After mapping several of these moduli, it will soon become obvious to the reader that every modulus represented in this way will have a set of unities, 1, -1, i, -i, and therefore, -1 will always be, at the least, a quadratic residue of these moduli, which, as mentioned earlier, is of the form p=4n+1, since all moduli of the form p=4n+1 can be represented in this square form. On the other hand, primes of the form p=4n+3, i.e. all primes not of the form p=4n+1, have no residues which satisfy the relation x^{4} º 1, and therefore, can not have an equivalent i and -i, nor the complex representation intrinsic to Ö{-1}. Thus, the answer to the query for the cases of moduli for which -1 is a quadratic residue, a question so central to the theory of quadratic reciprocity-link to pete's paper-, becomes immediately evident when these numbers are viewed from the standpoint of the complex plane.

Residual Classes

In the first treatise on biquadratic residues published by Gauss, he presents a classification of all residues into four classes: Class A contains all biquadratic residues. Class C contains quadratic residues that are biquadratic non-residues. Classes B and D both contain quadratic non-residues.

For modulus 5, the separation into 4 classes was simple; one of each of the four residues fell in each class. In general, however, he states that for moduli of the form 4n+1, there will be [(p-1)/4] terms in each class. In other words, every class will have the same number of terms, which means that there will always be the same number of quadratic residues which are biquadratic residues as there are biquadratic residues, and there will be twice the quadratic non-residues. How could this be? Always?

For modulus 5, the classifications are

Next to each residue is a letter, A, B, C, D, indicating which class each residue belongs to.

As mentioned before, the properties of 1, 2, 3, 4, are that of 1, i, -i, -1, respectively, or 1, -i, i, -1, respectively. For this modulus, the algebraic properties of the classes are easily seen: anything times 1 º itself (xa º a), i ·i º-1, (bb º c) etc., where a, b, c, d, are numbers from classes A, B, C, D, respectively. However, Gauss makes the bold claim that the algebraic properties that hold for the classes of modulus 5 hold for all numbers of their respective classes in all moduli of the form 4n+1!

For example, for p = 13 the classes are

A

1,

3,

9

B

2,

5,

6

C

4,

10,

12

D

7,

8,

11

Try it out. Multiply any number in class B with any number in class A and see if you get a number belonging to class B. Multiply any two numbers in class B and see if the result is a number belonging to class C, etc.

Furthermore, since the classes themselves act in the same way as the four unities 1, i, -i, -1 did in modulus 5, they themselves can be mapped onto the complex plane.

To begin an investigation of the significance of this form of classification, let us go back to the complex plane representation of these moduli.

Remember earlier, when the residues of modulus p=13 were organized by sets of associates, a beautiful pattern of three sets was formed, each with a biquadratic residue, a quadratic residue and two quadratic non-residues.

So far so good. Already, since each set has one term of each class, it is easily seen that each class must have an equal number of terms. Not only that, but this representation seems to make the placement of terms of classes B and D non-arbitrary, in the sense that, if B were to contain i, then the next term in each set moving clockwise from the biquadratic residue will give all the other quadratic non-residues that belong in class B. Conversely, the first terms counterclockwise from the biquadratic residues, that is, those which correspond to -i, would give all the terms of class D.

Can you tell which orientation is given by the table above?

Remember, numerically, the two above representations are the same, that is, they are both representations of modulus 13, and therefore the quality of each residue will be consistent. However, when placed in the four classes, there are two variations, a property that is only seen in the geometrical representation on the plane! Therefore, though Gauss does not mention any squares, or even complex numbers in his 1828 treatise, the use of the classes is yet another hint of Gauss' use of geometry to arrive at his many remarkable theorems, rather than the numbers and letters which line those pages.

To ask the tested question again: Do these properties hold for all moduli of the form 4n+1, as Gauss claims?

Here are some test subjects:

Modulus 17

Modulus 37

Modulus 41

Notice that the process of simply squaring each residue to get the biquadratic characteristics of the modulus is already becoming very hectic. What happens when larger numbers, such as 73 or 97 are investigated?!

Wait a minute.

The properties of the very next prime modulus after 13, i.e modulus 17, do not seem to follow any of the patterns related above! For instance, the whole class A is populated entirely by 1 and its associates! Does the algebraic properties of the classes still hold?

If so, how are the classes B and D defined? It seems as if there are no clear distinctions between the two associated sets of quadratic non-residues, relative to classes A and C, as there had been in the case of modulus 13.

It seems the tables have turned. Is there an ordering in the numbers that cannot be seen in the geometry?

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On 06 Mar 2008, 17:16.