Robert “Dr. Bob” Gardner

ETSU Abstract Algebra Club

Spring 2017

Study of the Quaternions

http://faculty.etsu.edu/gardnerr/5410/notes/Quaternions-Algebraic-Supplement.pdf

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:

The Quaternion Group

Small Groups

So the quaternion group is the smallest group that does not fall into a familiar category.

Hungerford’s Definition

This is introduced in Exercise I.2.3 on page 33, where the exercise is to show that the group is of order 8.

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Hungerford Changes Notation

1

i

k

j

-1

-j

-i

-k

Multiplication on the right by i is represented by a blue arrow.

Multiplication on the right by j is represented by a red arrow.

I give this as an example of the use of Cayley digraphs in Introduction to Modern Algebra (Section I.7 of Fraleigh).

Normal Subgroups of the Quarternion Group 1

Normal Subgroups of the Quarternion Group 2

Subgroups of the Quarternion Group

For Fans of Galois Theory

This is Exercise 27 on page 584 of Dummit and Foote’s Abstract Algebra, 3^rd Edition. It is an exercise with 6 parts.

The Quaternions

The Quarternions: A Prequel

Rings

Examples of Rings

Zero Divisors

My Hero, Zero!

Examples of Zero Divisors

The Quaternions: Definition

The Quaternions: An Observation

The Complex Numbers: Algebra

The Complex Numbers: Geometry

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).

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William Rowan Hamilton

In a letter he wrote late in his life to his son Archibald Henry, Hamilton tells the story of his discovery:

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“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 through and work by myself…

So the exact date of the birth of the quaternions is October 16, 1843.

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This quote is based on Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire, John Henry Press (2006).

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Division Ring

The Factor Theorem

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:

So far, so good!

No Surprise!

I don’t like the sound of that!

Surprise!

Infinite Number of Roots for a Quadratic!

The Factor Theorem and Algebraic Closure in the Quaternions

Sources

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Tests in Mathematics #131, Springer-Verlag (1991).

G. Gentili and D. C. Struppa, A New Theory of Regular Functions of a Quaternionic Variable, Advances in Mathematics 216 (2007), 279-301.

Two Dimensional Spheres

Without Commutivity…

Quaternionic Polynomials

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.

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), 379-389. An easier proof is given in Gentili and Struppa.

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.

Divisors in a Ring

A One-Sided Factor Theorem

Algebraic Closure for Fields

Algebraic Closure for Division Rings

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), 654-661.

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), 379-389.

How Many Roots – The Proof

What to Make of This… Fundamentally

References

1. D.S. Dummit and R.M. Foote’s Abstract Algebra, 3^rd Edition, Hoboken, NJ: John Wiley & Sons (2004).
2. J.B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Boston: Addison-Wesley (Pearson Education) (2002).
3. G. Gentili and D.C. Struppa, A New Theory of Regular Functions of a Quaternionic Variable, Advances in Mathematics 216 (2007), 279-301.
4. T.W. Hungerford, Algebra, NY: Springer-Verlag (1974).
5. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Tests in Mathematics #131, Springer-Verlag (1991).
6. I. Nivens’ “Equations in Quaternions,” American Mathematical Monthly, 48 (1941), 654-661.
7. A. Pogorui and M. V. Shapiro, “On the Structure of the Set of Zeros of Quaternionic Polynomials,” Complex Variables 49(6) (2004), 379-389.

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