A finite field with four elements

 In the last post, we discovered that for any prime p, ℤp is a finite field, where the entire set under addition is an abelian group and ℤp-{0} is an abelian group under multiplication. On the other hand, if n is a composite number, ℤn-{0} is not a group under multiplication because there will be at least two numbers between 1 and n-1 that multiply to n, which in modulo n arithmetic means j*k = 0, and neither j or k will have multiplicative inverses.

Let us instead look at an additive group of four elements, 0, 1, a and 1+a. Here is the Cayley table for addition.

 

      0    1    a    1+a

0   | 0    1    a    1+a

1   | 1    0    1+a   a 

a   | a    1+a   0    1 

1+a | 1+a  a     1    0 

Every element is its own additive inverse, so this is isomorphic to the Klein four-group. What about the non-zero elements under multiplication? There are three such elements, so if this is a group, it must be isomorphic to ℤ3. Let's figure out the Cayley table.

 

      1    a    1+a

1   | 1    a    1+a

a   | a    ?    ?  

1+a | 1+a  ?    ?  

Recall that every element must show up in every row and column, so a*(1+a) must equal 1, and likewise (1+a)*a.

      1    a    1+a

1   | 1    a    1+a

a   | a    ?    1  

1+a | 1+a  1    ?  

The other two question marks can be sussed out by using the Latin square rule.

      1    a    1+a

1   | 1    a    1+a

a   | a    1+a  1  

1+a | 1+a  1    a  

In regular multiplication, (1+a)²= 1 + 2a + a², but because this group uses mod 2 arithmetic, this simplifies to 1 + a². On the table we see a² = 1+a, so we get 1+ 1+a, and mod 2 simplifies this to a. I leave it as an exercise for the reader to see that any combination of these four elements, will be consistent with our addition and multiplication tables.

 

In the literature, a field whose order is q = pⁿ is called Fq or 𝔽q or GF(q), which stands for Galois field.

 

The Quaternion group

 The concept of a square root of a negative number dates back to ancient Greece, and is credited to Heron, whose name is immortalized in the method for finding the area of a triangle given the lengths of the three sides. The name imaginary number was coined by Rene Descartes in the 1600s, who did not like the concept and hoped he could kill it by mocking it. The concept did not die, and about a century later, Leonhard Euler became the first to use the lowercase letter i as the square root of -1.

 

Numbers of the form a + bi, where a and b are real numbers constitute the complex numbers C. The common way to represent the complex numbers as a picture is the complex plane, where the real numbers are represented by the x-axis and the imaginary numbers are lined up on the y-axis. The axes intersect at zero.

 

Just like the real numbers R, the complex numbers C are an abelian group under addition and C - {0} is an abelian group under multiplication. We have discussed that ({1, -1}, ✕) is a finite abelian group and a subgroup of the real numbers. Similarly, ({1, i, -1, -i}, ✕) is a finite abelian group and a subgroup of the complex numbers. Note that -i is also a square root of -1, so it has two square roots in much the same way any positive number has two square roots. For example, 3²=9 and (-3)²=9.

 

Enter William Rowan Hamilton (1805-1865), widely regarded as the greatest Irish mathematician of all time. He created a mathematical system he called the quaternions, which has not two square roots of -1, but six. The general form of a number in the quaternion system is a + bi + cj + dk, where a, b, c and d  are arbitrary real numbers and i, j and k are square roots of -1, as are -i, -j and -k.

 

The quaternion system under multiplication is non-abelian. Here are the rules for multiplying i, j and k.

 

ij = k      jk = i     ki = j

ji = -k     kj = -i    ik = -j

 

Here is the full Cayley table representation, where we can find ij in the square corresponding to row i, column j.

There are three subgroups of order 4, and since 8/2 = 4, these are all normal subgroups.

Here are the subgroups.

{1, i, -1, -i}

{1, j, -1, -j}

{1, k, -1, -k}

 

Since 8 is divisible by 8, 4, 2 and 1, we might expect there is a subgroup of order 2, and {1, -1} fills the bill.

The quaterions are a non-abelian group where all subgroups are normal. This cannot be said for any other non-abelian group we have studied.

I present group theory in terms of pure mathematics, but groups play a vital role in the study of modern physics, including the quaternions. This is a link to the Wikipedia page that tells more.

Commentary

I mentioned four mathematicians today, so let me present links to their biographies. Far be it for me to talk smack about any of these guys, but the general public is probably better acquainted with Rene Descartes than nay of the other names, because Descartes was also a philosopher. Very few mathematicians after the Greeks are famous for just their mathematical work. If we judged this quartet solely on their mathematics, Leonhard Euler (1707-1783) is the true giant. For the quantity and quality of his work, he could be compared to the composer Johann Sebastian Bach (1685-1750).

Link to the biography of William Rowan Hamilton.

 

Link to the biography of Leonhard Euler.

 

Link to the biography of Heron of Alexandria.

Link to the biography of Rene Descartes.

Cayley tables in a special form and nxn permutation matrices representing a group with n elements

To begin, let me introduce the Kronecker delta function, 𝞭(x,y), where x and y are both elements of some set S.

If two things are the same, the delta function returns a 1. If not, the function returns a zero. 

 

Easy peasy, lemon squeezy.

 

Next, a Cayley table in special form. So far, the top row and left most column were transposes of each other, the same group element in position k of the top row as in position k of the leftmost column.

 

This time, we will line up the group elements with their inverses, so if x is in position k in the top row, position k in the leftmost column will be x⁻¹.

Now, we will do the Kronecker delta of the Cayley table with the identity element (1).

   

This is what the identity matrix should look like, ones along the mail diagonal, zeros everyplace else, which is to say we use the Kronecker delta on i and j for every entry mij in matrix M.

 

Remarkably enough, if we do the Kronecker delta on our Cayley table with any element in S3, we will get the 6x6 matrix that represents that element.

Matrix multiplication agrees with the combination operator of the permutations. More than that, if we want to turn the identity matrix into an even permutation mtarix such as (123) or (132), it will take an even number of row swaps or column swaps. To change the identity into (12), (13) or (23), we will do an odd number of column swaps. 

 

The biography of Leopold Kronecker is remarkable. An excellent student, he was also from a wealthy family and had little interest in teaching, only research at the top level. He was finally persuaded to teach at the age of 39, but students found his lectures hard to follow.

 

His most famous quote is "God made the integers, all the rest is the work of man." He had a visceral hatred of the new concepts of infinity presented by Georg Cantor, and did not like the definition of any irrational real number being defined as the limit of some infinite series of rational numbers. It should be noted that Gauss, born long before Kronecker and Cantor, did do research into infinity and decided not to publish, fearing it would just cause controvesry.

 

The next post will be about invariants in conjugacy classes of permutation matrices.