Generalization with purpose

Modern number theory begins with the integers, all those whole numbers, positive, negative, and zero. A few times, when teaching number theory, I began simultaneously with the integers, the Gaussian integers, and the Eisenstein integers. One pedagogical reason is this: if you spend the first few weeks of number theory class proving things that students “learned” by fifth grade, then students will not likely be excited. The Euclidean algorithm can excite students, but proving the existence and uniqueness of prime decomposition is — from the students’ perspective — proving something they already (think they) know. Number theory teachers can get flustered by this. “No!” they say, “You only think you know this! It needs to be proven. It’s really not trivial! Really! Pay attention!”

I think there are two important roles for the instructor at this stage. One is to guide the student to wiser ignorance, as in Plato’s Meno, where Socrates says “Is he not better off in knowing his ignorance?”.  To this end, the instructor gives counterexamples, where irreducible elements $p$ (those that cannot be factored further) are not prime (do not satisfy the implication $p \vert ab \Rightarrow p \vert a$ or $p \vert b$). The first counterexample that I learned was in the ring $Z[\sqrt{-5}]$ in which $(1 + \sqrt{-5}) \cdot (1 - \sqrt{-5}) = 2 \cdot 3$. The element $2$ is irreducible but not prime. I think this counterexample is natural to the number theorist and instructor — having a prior background in quadratic rings — but most unusual to the elementary number theory student.  It also leaves the instructor having to convince the student that $1 \pm \sqrt{-5}$ and $2$ and $3$ are irreducible.  That is probably too difficult for the beginning student.  A more elementary example occurs in Hilbert’s monoid $\{ 1,5,9,13,17,\ldots \}$ of natural numbers congruent to one modulo $4$. In this monoid (under multiplication), 9 is irreducible. On the other hand, $9 \vert 21 \cdot 21$ but 9 does not divide 21. Another way to say this is that 441 factors into irreducible in two ways:  $9 \cdot 49$ and $21 \cdot 21$.  So this Hilbert monoid might break the student’s false intuition that irreducible elements are prime, and that factorization must be unique.

Besides leading the student to wiser ignorance, the instructor should motivate the proof of existence and uniqueness of prime decomposition by demonstrating its application to number systems beyond the integers.  To this end, one strategy is to simultaneously teach students the basic arithmetic of integers, Gaussian integers, and Eisenstein integers. I think students appreciate the details of proofs when they are proving something they don’t already “know”. Only those already devoted to pure mathematics will appreciate a proof of prior “knowledge”.

I have devoted Chapter 5 to a study of Gaussian and Eisenstein integers. I hope that some instructors will follow this chapter as they go through Chapters 2 and 3 to pique the interest of students. It is not technically necessary for what comes later, but Gaussian and Eisenstein integers are beautiful and important in their own right. I have always liked their kaleidoscopic symmetry, so I devoted an entire two-page spread to visualizations of Gaussian and Eisenstein primes.

The pictures display the ramified, inert, and split primes in different colors, and the inert primes are displayed below on a number line to provide a scale. A fundamental domain is highlighted to exhibit the kaleidoscopic symmetry. Symmetry will reappear in the study of orthogonal groups of binary quadratic forms later, so it is good to see a few examples here.  Hecke proved equidistribution of prime angles in the Gaussian context (and much more), in efforts to prove the infinitude of primes of the form $x^2 + 1$. Thus I have drawn attention to the prime angles (with tick marks on the circumference) and the primes of the form $x^2 + 1$ (with parallel horizontal lines across the Gaussian plane).  Analogously, I have drawn attention to the primes of the form $x^2 - x + 1$ in the diagram of Eisenstein integers.

Why else study Gaussian and Eisenstein integers?  Some students think that generalization is a worthy goal for its own sake, but I prefer generalization with purpose.  In Chapter 5, we use properties of Gaussian and Eisenstein primes to study properties of ordinary primes (infinitude of primes congruent to one modulo three and modulo four, and relations to primes of the form $x^2 + 1$).  One “big picture” lesson of this book is that the study of larger and stranger number systems sheds light on the natural numbers.  Exhibit G:  The Gaussian Integers.  Exhibit E:  The Eisenstein Integers.  Later on, quadratic rings, and the rings of modular arithmetic.

Why is the treatment of Gaussian and Eisenstein integers delayed until Chapter 5, instead of following my pedagogical advice to include them in the teaching of prime decomposition?  There are two reasons.  First, for shorter courses, I can imagine an instructor skipping Chapter 5.  Second, Gaussian and Eisenstein integers connect the first part of the book (foundational properties of integers and rational numbers) to the second part of the book (on binary quadratic forms).  By Chapter 7, we will study definite binary quadratic forms, and the Gaussian and Eisenstein integers will reappear in relation to the unique forms of discriminants $-3$ and $-4$, respectively.