1. The Quaternions: An Algebraic Approach Robert “Dr. Bob” Gardner1 i j k -1 -i -j -k
The Quaternions:
An Algebraic Approach
Robert “Dr. Bob” Gardner
ETSU Abstract Algebra Club
Fall 2018

2. Study of the Quaternions
Geometrically: The quaternions, like the complex numbers, can be used to perform rotations in 3 or 4 dimensions.
Analytically: A new analytic theory of functions of a “quaternionic variable” has recently been developed. Many results parallel those of complex analytic funtions.
Algebraically: Here we introduce the quaternion group of order 8 and the quaternions as a noncommutative division ring. We present a Fundamental Theorem of Algebra for Quaternions and describe the structure of the set of roots of a polynomial.
The proofs of the results are online in my notes for Modern Algebra 2 (MATH 5420) in the chapter on rings:

3. The Quaternion Group

4. Small Groups
ORDER
GROUPS
COMMENTS 1
ℤ 1
Trivial Group 2
ℤ 2 3
ℤ 3 ≅ 𝐴 3 4
ℤ 4
𝑉≅ ℤ 2 × ℤ 2
Klein-4, smallest noncyclic group 5
ℤ 5 6
ℤ 6 ≅ ℤ 2 × ℤ 3
𝑆 3 ≅ 𝐷 3
Smallest nonabelian group 7
ℤ 7 8
ℤ 8
ℤ 2 × ℤ 4
ℤ 2 × ℤ 2 × ℤ 2
𝐷 4
𝑄 8
Nonabelian
Quaternion group, nonabelian
So the quaternion group is the smallest group that does not fall into a familiar category.

5. Hungerford’s Definition
Hungerford defines the quaternion group in his Section I.2 (Homo-morphisms and Subgroups) as the multiplicative group generated by the matrices:
𝐴= 0 1 −1 0 and B= 0 𝑖 𝑖 0 .
This is introduced in Exercise I.2.3 on page 33, where the exercise is to show that the group is of order 8.

6. Hungerford Changes Notation
In Exercise II.6.3 Hungerford considers the set ±1, ±𝑖, ±𝑗, ±𝑘 with multiplication given by the equations 𝑖 2 = 𝑗 2 = 𝑘 2 =−1, 𝑖𝑗=𝑘, 𝑗𝑘=𝑖, 𝑘𝑖=𝑗, 𝑗𝑖=−𝑘, 𝑘𝑗=−𝑖, 𝑖𝑘=−𝑗, and the usual rules for multiplying by ±1. The exercise is to show that the resulting group is isomorphic to the quaternion group.

7. Cayley Digraph for 𝑄 8 1 i j k -k -j -i -1
I give this as an example of the use of Cayley digraphs in Introduction to Modern Algebra (Section I.7 of Fraleigh). k -k 1 i j k -1 -i -j -k -j -i -1
Multiplication on the right by i is represented by a blue arrow.
Multiplication on the right by j is represented by a red arrow.

8. An Easier Diagram for the 𝑄 8 Table𝑖
𝑘𝑖=𝑗
𝑖𝑗=𝑘 𝑗 𝑘
𝑗𝑘=𝑖

9. An Easier Diagram for the 𝑄 8 Table𝑖
𝑖𝑘=−𝑗
𝑗𝑖=−𝑘 𝑗 𝑘
𝑘𝑗=−𝑖
In fact, cross products of the vectors 𝑖 , 𝑗 , 𝑘 ∈ ℝ 3 behave exactly like this under cross products:
𝑖 × 𝑗 = 𝑘 =− 𝑗 × 𝑖 , 𝑗 × 𝑘 = 𝑖 =− 𝑘 × 𝑗 , 𝑘 × 𝑖 = 𝑗 =− 𝑖 × 𝑘 .

10. Normal Subgroups of the Quarternion Group 1
In Hungerford’s Exercise I.5.8, it is shown that every subgroup of 𝑄 8 is normal. By Lagrange’s Theorem (Fraleigh’s Theorem 10.10, Hungerford’s Corollary I.4.6), the only possible orders of subgroups are 1, 2, 4, and 8. Trivially, the improper groups of order 1 and 8 are normal. Every subgroup of index 2 (that is, of order half the size of the larger group… assuming finite groups) is a normal subgroup by Fraleigh’s Exercise and Hungerford’s Exercise I.5.1, so the subgroups of order 4 must be normal. Now for subgroups of order 2, we must find an element of order 2 in 𝑄 8 …

11. Normal Subgroups of the Quarternion Group 2
The only element of 𝑄 8 of order 2 is −1, so the only subgroup of order 2 is 𝑁={1, −1}. For any 𝑞∈𝑄 8 , we “clearly” have 𝑞 1 𝑞 −1 =1.
Less clear, but also true is that for any 𝑞∈𝑄 8 we have 𝑞 −1 𝑞 −1 =−1. So for any 𝑞∈𝑄 8 we have 𝑞 𝑁 𝑞 −1 ⊂𝑁; hence 𝑁 is a normal subgroup. Ergo(!), all subgroups of 𝑄 8 are normal. 1 i j k -1 -i -j -k

12. Subgroups of the Quarternion Group
We just established that every subgroup of 𝑄 8 is normal, without actually finding the subgroups (except for one)! We find that the subgroup diagram of 𝑄 8 is:
𝑄 8
1, −1, 𝑖, −𝑖
1, −1, 𝑗, −𝑗
1, −1, 𝑘, −𝑘
1, −1 1

13. For Fans of Galois Theory
The quaternion group, 𝑄 8 , is the Galois group Aut ℚ ℚ 𝛼 where 𝛼= That is, the group of automorphisms of the algebraic extension of the rationals, ℚ, by 𝛼 is (isomorphic to) 𝑄 8 . The irreducible polynomial for 𝛼 is 𝑥 8 −24 𝑥 𝑥 4 −288 𝑥
This is Exercise 27 on page 584 of Dummit and Foote’s Abstract Algebra, 3rd Edition. It is an exercise with 6 parts.

14. The Quaternions

15. The Quarternions: A Prequel
The quaternions are based on writing linear combinations (with real coefficients) of the elements in 𝑄 8 and then adding and multiplying them. This requires more than just a group structure, though. 1 i j k -1 -i -j -k

16. Rings
Definition (Hungerford’s III.1.1). A ring is a nonempty set 𝑅 together with two binary operations (denoted + and multiplication) such that
(i) 𝑅,+ is an abelian group.
(ii) 𝑎𝑏 𝑐=𝑎 𝑏𝑐 for all 𝑎, 𝑏, 𝑐 in 𝑅 (i.e., multiplication is associative).
(iii) 𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐 and 𝑎+𝑏 𝑐=𝑎𝑐+𝑏𝑐 (left and right distribution of multiplication over +).
If in addition,
(iv) 𝑎𝑏=𝑏𝑎 for 𝑎,𝑏 in 𝑅
then 𝑅 is a commutative ring. If 𝑅 contains an element 1 𝑅 such that
(v) 1 𝑅 𝑎=𝑎 1 𝑅 =𝑎 for all 𝑎 in 𝑅,
then 𝑅 is a ring with identity (or unity).

17. Examples of Rings You are familiar with the following rings:
ℂ,ℝ,ℚ, and ℤ
All 2×2 matrices with real entries, ℳ 2 ℝ
The rings ℂ,ℝ,ℚ, and ℤ are commutative rings with unity. ℳ 2 ℝ is a ring with unity, but it is not commutative. The ring 2ℤ is a commutative ring which does not have unity.

18. My Hero, Zero! Zero Divisors
Definition (Hungerford’s III.1.3). A nonzero element 𝑎 in the ring 𝑅 is a left (respectively, right) zero divisor if there exists nonzero 𝑏 in 𝑅 such that 𝑎𝑏=0 (respectively 𝑏𝑎=0). A zero divisor is an element of 𝑅 which is both a left and right zero divisor.
My Hero, Zero!

19. Examples of Zero Divisors
The integers modulo 𝑛, ℤ 𝑛 ≅ℤ/𝑛ℤ, forms a ring where multiplication of equivalence classes is defined in terms of multiplication of representative. If 𝑛 is a composite number (and only if, in fact), then ℤ 𝑛 has zero divisors. For example, 2 is a zero divisor in ℤ 6 since 2 × 3 = 6 = 0 .
In the ring ℳ 2 ℝ , there are zero divisors. For example
−1 −1 =
So is a left divisor of zero and −1 −1 is a right divisor of zero.

20. The Quaternions: Definition
Definition (Hungerford’s page 117). Let 𝑆= 1,𝑖,𝑗,𝑘 . Let ℍ be the additive abelian group ℝ⨁ℝ⨁ℝ⨁ℝ and write the elements of ℍ as formal sums 𝑎 0 , 𝑎 1 , 𝑎 2 , 𝑎 3 = 𝑎 0 1+ 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘. We often drop the “1” in “ 𝑎 0 1” and replace it with just 𝑎 0 . Addition in ℍ is then:
𝑎 0 + 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘 + 𝑏 0 + 𝑏 1 𝑖+ 𝑏 2 𝑗+ 𝑏 3 𝑘 = 𝑎 0 + 𝑏 𝑎 1 + 𝑏 1 𝑖+ 𝑎 2 + 𝑏 2 𝑗+ 𝑎 3 + 𝑏 3 𝑘.
We turn ℍ into a ring by defining multiplication as
𝑎 0 + 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘 𝑏 0 + 𝑏 1 𝑖+ 𝑏 2 𝑗+ 𝑏 3 𝑘 = 𝑎 0 𝑏 0 − 𝑎 1 𝑏 1 − 𝑎 2 𝑏 2 − 𝑎 3 𝑏 3 𝑎 0 𝑏 0 − 𝑎 1 𝑏 1 − 𝑎 2 𝑏 2 − 𝑎 3 𝑏 𝑎 0 𝑏 1 + 𝑎 1 𝑏 0 + 𝑎 2 𝑏 3 − 𝑎 3 𝑏 2 𝑖+ 𝑎 0 𝑏 2 + 𝑎 2 𝑏 0 + 𝑎 3 𝑏 1 − 𝑎 1 𝑏 3 𝑗+ 𝑎 0 𝑏 3 + 𝑎 3 𝑏 0 + 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 𝑎 0 𝑏 3 + 𝑎 3 𝑏 0 + 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 𝑘.

21. The Quaternions: An Observation
The product above can be interpreted by considering:
(i) multiplication in the formal sum is associative,
(ii) 𝑟𝑖=𝑖𝑟, 𝑟𝑗=𝑗𝑟, 𝑟𝑘=𝑘𝑟 for all 𝑟 in 𝑅,
(iii) 𝑖 2 = 𝑗 2 = 𝑘 2 =𝑖𝑗𝑘=−1, 𝑖𝑗=−𝑗𝑖=𝑘,
𝑗𝑘=−𝑘𝑗=𝑖, 𝑘𝑖=−𝑖𝑘=𝑗.
The resulting ring is the ring of real quaternions (we still need to verify associativity of multiplication and distribution) . The notation ℍ is in honor of Sir William Rowan Hamilton ( ) who discovered them in 1843.

22. The Complex Numbers: Algebra
The complex numbers can be defined as ordered pairs of real numbers,
ℂ= 𝑎,𝑏 𝑎,𝑏∈ℝ ,
with addition defined as
𝑎,𝑏 + 𝑐,𝑑 = 𝑎+𝑐,𝑏+𝑑
and multiplication defined as
𝑎,𝑏 𝑐,𝑑 = 𝑎𝑐−𝑏𝑑,𝑏𝑐+𝑎𝑑 .
We then have that ℂ is a field with additive identity 0,0 and multiplicative identity 1,0 . The additive inverse of 𝑎,𝑏 is −𝑎,−𝑏 and the multiplicative inverse of 𝑎,𝑏 ≠ 0,0 is 𝑎 𝑎 2 + 𝑏 2 , −𝑏 𝑎 2 + 𝑏 We commonly denote 𝑎,𝑏 as “𝑎+𝑖𝑏” so that 𝑖= 0,1 and we notice that 𝑖 2 =−1.

23. Graphing Complex Numbers
The complex numbers are visualized as the “complex plane” where 𝑎+𝑖𝑏∈ℂ is associated with 𝑎,𝑏 ∈ ℝ 2 .
Im 𝑧 𝑦 ℂ
ℝ 2
𝑎+𝑖𝑏
𝑎,𝑏 𝑏 𝑏
Re 𝑧 𝑥 𝑎 𝑎
During the early decades of the 19th century, the complex numbers became an accepted part of mathematics (in large part due to the development of complex function theory by Augustin Cauchy).

24. William Rowan Hamilton
Sir William Rowan Hamilton ( ) spent the years 1835 to 1843 trying to develop a three dimensional number system based on triples of real numbers. He never succeeded. However, he did succeed in developing a four dimensional number system, now called the quaternions and denoted “ℍ” in his honor.
In a letter he wrote late in his life to his son Archibald Henry, Hamilton tells the story of his discovery:

25. So the exact date of the birth of the quaternions is October 16, 1843.
“Every morning in the early part of [October 1843], on my coming down to breakfast, your little brother, William Edwin, and yourself, used to ask me, ‘Well, papa, can you multiply triplets?’ Whereto I was always obliged to reply, with a sad shake of the head: ‘No, I can only add and subtract them.’ But on the 16th day of that same month… An electric circuit seemed to close; and a spark flashed forth the herald (as I foresaw immediately) of many long years to come of definitely directed thought and work by myself…
So the exact date of the birth of the quaternions is October 16, 1843.
This quote is based on Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire, John Henry Press (2006).

26. … Nor could I resist the impulse-unphilosophical as it may have been-to cut with a knife on a stone of Brougham Bridge [in Dublin, Ireland; now called “Broom Bridge”], as we passed it, the fundamental formula with the symbols 𝑖, 𝑗, 𝑘: 𝑖 2 = 𝑗 2 = 𝑘 2 =𝑖𝑗𝑘=−1 which contains the Solution of the Problem, but, of course, the inscription has long wince mouldered away.”

27. 𝑞=0 Graphing Quaternions Im 𝑞 Re 𝑞 +1 1 1 1 −1
Just as we can graph complex numbers using two real axes, we can graph quaternions using four real axes. Since we can’t visualize this, we pull one copy of the real axis off to the left (for the “real part” of quaternion 𝑞) and leave the other three axes to represent the “imaginary parts” of 𝑞 (also called the “vector part”).
Im 𝑞
Re 𝑞 +1 1 1 −1 1
𝑞=0

28. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞= 1 2 +𝑖 1 2 +𝑗 1 2 +𝑘 1 2

29. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞∈1+0S
The Unit Ball

30. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞∈ S
The Unit Ball

31. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞∈0+1S
The Unit Ball

32. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞∈− S
The Unit Ball

33. Im 𝑞
Re 𝑞 +1 1 1 1 −1
𝑞∈−1+0S
The Unit Ball

34. The Quaternions as a Noncommutative Division Ring

35. Division Ring
Definition (Hungerford’s III.1.5). A commutative ring 𝑅 with (multiplicative) identity 1 𝑅 and no zero divisors is an integral domain. A ring 𝐷 with identity 1 𝐷 ≠0 in which every nonzero element is a unit is a division ring. A field is a commutative division ring.
Note. In ℍ we have the identity 1= 1,0,0,0 . Since 𝑖𝑗=−𝑗𝑖≠𝑖𝑗, then ℍ is not commutative and so ℍ is not an integral domain nor a field.
Theorem. The quaternions ℍ form a noncommutative division ring.

36. ℍ is a Noncommutative Division Ring
Theorem. The quaternions ℍ form a noncommutative division ring.
Proof. Extremely tedious computations confirm that multiplication is associative and that the distribution laws hold. We now show that every nonzero element of ℍ has a multiplicative inverse. Consider q= 𝑎 0 + 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘. Define 𝑑= 𝑎 𝑎 𝑎 𝑎 3 2 ≠0. Notice that:
𝑎 0 + 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘 (𝑎 0 𝑑) − ( 𝑎 1 𝑑) 𝑖− (𝑎 2 𝑑) 𝑗− ( 𝑎 3 𝑑 )𝑘 = 𝑎 0 −𝑎 0 /𝑑 − 𝑎 1 − 𝑎 1 𝑑 − 𝑎 2 − 𝑎 2 𝑑 − 𝑎 3 − 𝑎 3 𝑑 + 𝑎 0 −𝑎 1 /𝑑 + 𝑎 1 𝑎 0 𝑑 + 𝑎 2 − 𝑎 3 𝑑 − 𝑎 3 − 𝑎 2 𝑑 𝑖+ 𝑎 0 −𝑎 2 /𝑑 + 𝑎 2 𝑎 0 𝑑 + 𝑎 3 − 𝑎 1 𝑑 − 𝑎 1 − 𝑎 3 𝑑 𝑗+ 𝑎 0 −𝑎 3 /𝑑 + 𝑎 3 𝑎 0 𝑑 + 𝑎 1 − 𝑎 2 𝑑 − 𝑎 2 − 𝑎 1 𝑑 𝑘= 𝑎 𝑎 𝑎 𝑎 𝑑=1.

37. ℍ is a Noncommutative Division Ring (continued)
Theorem. The quaternions ℍ form a noncommutative division ring.
Proof (continued). So 𝑞 −1 = 𝑎 0 + 𝑎 1 𝑖+ 𝑎 2 𝑗+ 𝑎 3 𝑘 −1 = 𝑎 0 𝑑 − 𝑎 1 𝑑 𝑖− 𝑎 2 𝑑 𝑗− 𝑎 3 𝑑 𝑘 where 𝑑= 𝑎 𝑎 𝑎 𝑎 Therefore every nonzero element of ℍ is a unit and so the quaternions form a noncommutative division ring. ∎
Note. Since every nonzero element of ℍ is a unit, then ℍ contains no left zero divisors: If 𝑞 1 𝑞 2 =0 and 𝑞 1 ≠0, then 𝑞 2 = 𝑞 1 −1 0=0. Similarly, ℍ has no right zero divisors.

38. The Number of Roots of a Quaternionic Polynomial

39. The Factor Theorem
The Factor Theorem (Hungerford’s Theorem III.6.6). Let 𝑅 be a commutative ring with identity and 𝑓∈𝑅[𝑥]. Then 𝑐∈𝑅 is a root of 𝑓 if and only if 𝑥−𝑐 divides 𝑓.
Since ℍ is not commutative, the Factor Theorem need not hold there.

40. A Bound on the Number of Roots in an Integral Domain
The Factor Theorem is used to prove the following, which should remind you of the Fundamental Theorem of Algebra:
Hungerford’s Theorem III If 𝐷 is an integral domain contained in an integral domain 𝐸 and a 𝑓∈𝐷[𝑥] has degree 𝑛, then 𝑓 has at most 𝑛 distinct roots in 𝐸.
So far, so good!

41. I don’t like the sound of that!
No Surprise!
So by the previous theorem, in an integral domain an 𝑛 degree polynomial has at most 𝑛 roots. You have probably dealt with polynomials over fields where there are not as many roots as the degree of the polynomial.
For example, polynomial 𝑥 2 −2 does not have any roots over field ℚ but has two roots over field ℝ. Polynomial 𝑥 3 −1 has one root over ℝ (and one root over ℚ), but has three roots over ℂ. But you have probably not encountered a situation where a polynomial has more roots than its degree.
I don’t like the sound of that!

42. Surprise!
Example. The polynomial 𝑞 2 +1∈ℍ 𝑞 (we use 𝑞 to represent a “quaternionic” variable) has more than two roots. Along with ±𝑖 there are ±𝑗 and ±𝑘. In fact, the polynomial has an infinite number of roots in ℍ!

43. Infinite Number of Roots for a Quadratic!
Example. Let 𝑥 1 , 𝑥 2 , 𝑥 3 ∈ℝ with 𝑥 𝑥 𝑥 =1. We have
𝑥 1 𝑖+ 𝑥 2 𝑗+ 𝑥 3 𝑘 2 = 𝑥 𝑖 2 + 𝑥 1 𝑥 2 𝑖𝑗+ 𝑥 1 𝑥 3 𝑖𝑘+ 𝑥 2 𝑥 1 𝑗𝑖+ 𝑥 𝑗 2
+ 𝑥 2 𝑥 3 𝑗𝑘+ 𝑥 3 𝑥 1 𝑘𝑖+ 𝑥 3 𝑥 2 𝑘𝑗+ 𝑥 𝑘 2
by the definition of multiplication
= − 𝑥 − 𝑥 − 𝑥 since 𝑖𝑗=−𝑗𝑖,
𝑖𝑘=−𝑘𝑖, 𝑗𝑘=−𝑘𝑗, 𝑖 2 = 𝑗 2 = 𝑘 2 =−1
=−1 since 𝑥 𝑥 𝑥 =1.
Therefore, 𝑥 1 𝑖+ 𝑥 2 𝑗+ 𝑥 3 𝑘 is a root of 𝑞 2 +1∈ℍ 𝑞 for all 𝑥 𝑥 𝑥 =1 and hence 𝑞 2 +1 has an infinite number of roots.

44. The Factor Theorem and Algebraic Closure in the Quaternions

45. Sources
T. Y. Lam, A First Course in Noncommutative Rings,2nd Edition, Graduate Tests in Mathematics #131, Springer-Verlag (2001).
G. Gentili and D. C. Struppa, A New Theory of Regular Functions of a Quaternionic Variable, Advances in Mathematics 216 (2007),

46. Two Dimensional Spheres
Definition. Denote by 𝕊 the two dimensional sphere (as a subset of the four dimensional quaternions ℍ), 𝕊= 𝑞= 𝑥 1 𝑖+ 𝑥 2 𝑗+ 𝑥 3 𝑘 𝑥 𝑥 𝑥 =1 .
For 𝑥, 𝑦∈ℝ we let 𝑥+𝑦𝕊 denote the two dimensional sphere 𝑥+𝑦𝕊= 𝑥+𝑦𝐼 𝐼∈𝕊 .
Note. We have seen that for any 𝐼∈𝕊 we have 𝐼 2 +1=0. We might think of 𝑥+𝑦𝕊 as a two dimensional sphere centered at (𝑥,0,0,0) with radius |𝑦|. Spheres play a role in the roots of quaternionic polynomials.

47. Without Commutivity…
If 𝑅 is a ring with identity, then by Hungerford’s Theorem III.5.2(ii), in the ring of polynomials 𝑅 𝑡 (with 𝑡 as the indeterminate) we have that all 𝑟∈𝑅 commute with 𝑡 so that 𝑛=0 𝑁 𝑟 𝑛 𝑡 𝑛 = 𝑛=0 𝑁 𝑡 𝑛 𝑟 𝑛 . But if 𝑅 is a noncommutative ring (such as ℍ), then there is ambiguity as to how to define the evaluation homomorphism.
For example, in ℍ 𝑞 the polynomials 𝑎 𝑞 𝑛 and 𝑎 0 𝑞 𝑎 1 𝑞⋯ 𝑎 𝑛 𝑞, where 𝑎= 𝑎 0 𝑎 1 ⋯ 𝑎 𝑛 , are the same, but for 𝑟∈ℍ we may not have that 𝑎 𝑟 𝑛 = 𝑎 0 𝑟 𝑎 1 𝑟⋯𝑟 𝑎 𝑛 𝑟.

48. Quaternionic Polynomials
Definition. We define a left quaternionic polynomial 𝑝 to be of the form 𝑝 𝑞 = 𝑛=0 𝑁 𝑞 𝑛 𝑎 𝑛 . A right quaternionic polynomial 𝑝 to be of the form 𝑝 𝑞 = 𝑛=0 𝑁 𝑎 𝑛 𝑞 𝑛 .
Note. We now have an unambiguous way to evaluate left and right quaternionic polynomials. However, they do not (yet) form a polynomial ring since it is not clear how to multiply them.
Definition. For two (left) quaternionic polynomials 𝑝 1 𝑞 = 𝑛=0 𝑁 𝑞 𝑛 𝑎 𝑛 and 𝑝 2 𝑞 = 𝑚=0 𝑀 𝑞 𝑚 𝑏 𝑚 define the product
( 𝑝 1 𝑝 2 ) 𝑞 = 𝑛=0, 1, …, 𝑁;𝑚=0,1,…,𝑀 𝑞 𝑛+𝑚 𝑎 𝑛 𝑏 𝑚 .

49. Two Certain Roots Imply Infinite Roots
The following result is originally due to A. Pogorui and M. V. Shapiro (in “On the Structure of the Set of Zeros of Quaternionic Polynomials,” Complex Variables 49(6) (2004), An easier proof is given in Gentili and Struppa.
Theorem. Let 𝑝 𝑞 = 𝑛=0 𝑁 𝑞 𝑛 𝑎 𝑛 be a left quaternionic polynomial. Suppose there exist 𝑥 0 , 𝑦 0 ∈ℝ and 𝐼,𝐽∈𝕊 with 𝐼≠𝐽 such that 𝑝 𝑥 0 + 𝑦 0 𝐼 =0 and 𝑝 𝑥 0 + 𝑦 0 𝐽 =0. Then for all 𝐿∈𝕊, we have 𝑝 𝑥 0 + 𝑦 0 𝐿 =0.
The proof is fairly straightforward and computational. Gentili and Struppa develop a theory of analytic functions of a quaternionic variable and show that this theorem holds for analytic functions.

50. Divisors in a Ring
Note. The absence of commutivity in ℍ (and in rings in general) has an effect on the concept of a divisor, as we saw foreshadowed when dealing with divisors of zero.
Definition. Let 𝑅 be a ring. For 𝑟∈𝑅, if there are 𝑎, 𝑏∈𝑅 where 𝑟=𝑎𝑏, then 𝑎 is a left divisor of 𝑟 and 𝑏 is a right divisor of 𝑟.

51. A One-Sided Factor Theorem
Definition 16.1 of Lam. Let 𝑅 be a ring and 𝑝 𝑡 = 𝑛=0 𝑁 𝑡 𝑛 𝑎 𝑛 ∈𝑅 𝑡 . An element 𝑟∈𝑅 is a left root of 𝑝 if 𝑝 𝑞 = 𝑛=0 𝑁 𝑟 𝑛 𝑎 𝑛 =0. A right root is similarly defined.
Proposition 16.2 of Lam (The Factor Theorem in a Ring with Unity).
Let 𝑅 be a ring with unity. An element 𝑟∈𝑅 is a left root of nonzero polynomial 𝑝 𝑡 = 𝑛=0 𝑁 𝑡 𝑛 𝑎 𝑛 ∈𝑅 𝑡 if and only if 𝑡−𝑟 is a left divisor of 𝑝(𝑡)∈𝑅 𝑡 . The same holds for right roots and right divisors.

52. Algebraic Closure for Fields
Definition (Hungerford’s Section V.3). A field 𝐹 is algebraically closed if every nonconstant polynomial 𝑝∈𝐹 𝑥 has a root in 𝐹.
Note. By the Factor Theorem (and induction), a polynomial of degree 𝑛 in an algebraically closed field 𝐹 can be written as a product of linear terms in 𝐹 𝑥 :
𝑝 𝑥 = 𝑛=0 𝑁 𝑎 𝑛 𝑥 𝑛 = 𝑎 𝑁 𝑛=1 𝑁 𝑥− 𝑟 𝑛 .
FTA: ℂ is algebraically closed.

53. Algebraic Closure for Division Rings
Definition (Lam, page 255). A division ring 𝐷 is left (right) algebraically closed if every nonconstant polynomial in 𝐷 𝑡 has a left (right) root in 𝐷.
Note. The following is the Fundamental Theorem of Algebra for Quaternions. The result originally appeared in I. Niven’s “Equations in Quaternions,” American Mathematical Monthly, 48 (1941),
Theorem of Lam (“Niven-Jacobson” in Lam) Fundamental Theorem of Algebra for Quaternions.
The quaternions, ℍ, are left (and right) algebraically closed.

54. But, How Many Roots?
Note. The following result is from A. Pogorui and M. Shapiro’s “On the Structure of the Set of Zeros of Quaternionic Polynomials,” Complex Variables : Theory and Applications 49(6) (2004),
Theorem (Pogorui and Shapiro). Let 𝑝 be a nonzero polynomial of degree 𝑛 in ℍ 𝑞 . The set of left (right) roots of 𝑝 consists of isolated points or isolated two dimensional spheres of the form 𝑆=𝑥+𝑦𝕊 for 𝑥,𝑦∈ℝ. The number of isolated roots plus twice the number of isolated spheres is less than or equal to 𝑛.

55. How Many Roots – The Proof
The Proof of Pogoriu and Shapiro’s Theorem is based on introducing a polynomial of degree 2𝑛 with real coefficients (called the “basic polynomial”) which is associated with a given quaternionic polynomial of degree 𝑛. A one to one correspondence between the isolated zeros of the quaternionic polynomial and the basic polynomial is established, and a one to one correspondence between the isolated sphere of roots of the quaternionic polynomial and pairs of complex conjugate roots of the basic polynomial is established. Then the fact that a real polynomial of degree 2𝑛 has at most 2𝑛 complex roots (by the Fundamental Theorem of Algebra) is used to complete the proof.

56. The Real Numbers, the Complex Numbers, or the Quaternions?

57. What to Make of This… Fundamentally
“My interpretation of this, is that the cleanest place to address polynomials and algebraic closure is in the complex field ℂ! The real field ℝ is too small (it doesn’t even have roots for 𝑥 2 +1), the quaternions ℍ are too exotic and weird ( 𝑥 2 +1 has infinitely many roots!), but ℂ is just right!”

58. References
D.S. Dummit and R.M. Foote’s Abstract Algebra, 3rd Edition, Hoboken, NJ: John Wiley & Sons (2004).
J.B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Boston: Addison-Wesley (Pearson Education) (2002).
G. Gentili and D.C. Struppa, A New Theory of Regular Functions of a Quaternionic Variable, Advances in Mathematics 216 (2007),
T.W. Hungerford, Algebra, NY: Springer-Verlag (1974).
T. Y. Lam, A First Course in Noncommutative Rings, 2nd Edition, Graduate Tests in Mathematics #131, Springer-Verlag (2001).
I. Nivens’ “Equations in Quaternions,” American Mathematical Monthly, 48 (1941),
A. Pogorui and M. V. Shapiro, “On the Structure of the Set of Zeros of Quaternionic Polynomials,” Complex Variables 49(6) (2004),