Today in my Abstract Algebra class, the teacher talked about division rings and provided an example of a non-commutative division ring, namely the ring of quaternions. The teacher then mentioned that “there are only four types of finite-dimensional division algebras over the real field,” which are the real numbers themselves, complex numbers, quaternions, and octonions (here, division algebras refer to division rings). This statement sounded familiar to me, but something felt slightly off. I remembered reading in a book that if a system is defined as a hypercomplex number system over the reals and satisfies the multiplicativity of the norm, then there are only these four types. However, the teacher’s statement implied that even if the multiplicativity of the norm is removed, there are still only four. Naturally, I thought the teacher might have misremembered and debated with him for a while. After returning to my dormitory and looking up information, I finally confirmed: There really are only four finite-dimensional division algebras over the real field! Below is a brief discussion of my understanding of this issue.
Of course, it is impossible to provide a formal proof of this proposition here, as the proof is quite complex and I have not yet fully grasped it myself. However, it is possible to get a rough sense of why this is the case. Upon seeing this proposition, our immediate reaction might be: How can there be so few! We will use examples to briefly illustrate that there indeed cannot be many!
We are already very familiar with the complex number system, which is a vector space defined over the reals with the basis \{1, i\}, and the multiplication is given by: 1 \times i = i \times 1 = i, \quad 1^2 = 1, \quad i^2 = -1 Then division can be defined. We eventually find that, except for 0, all elements in this vector space have a unique inverse, which is a necessary condition for a division ring! Another vector space very similar to the complex numbers has the basis \{1, j\}, but defines j^2 = 1. However, this is not a division ring!
Where did it go wrong? Let us consider the inverse of a + bj, where a and b are not both zero: \frac{1}{a+bj} = \frac{a-bj}{(a+bj)(a-bj)} = \frac{a-bj}{a^2-b^2} Do you see the denominator? In the case of complex numbers, this denominator is a^2 + b^2, which is non-zero for any a, b that are not both zero. But here, the denominator is a^2 - b^2. Even if a and b are not both zero, it is still possible for the denominator to be zero, in which case the inverse cannot be defined. In other words, certain non-zero elements in this ring do not have an inverse, which prevents it from being a division ring.
Looking at the difference between a^2 + b^2 and a^2 - b^2, we can see that the former is positive definite while the latter is not. Homogeneous positive definiteness ensures that as long as the variables are not all zero, the result will not be zero. This seems to be the core difference between the two. If this is truly the case, then we need to find the significance of a^2 + b^2 and a^2 - b^2. Let’s look at finding the inverse from another perspective. Suppose the inverse of a + bj is c + dj, then: 1 = (a + bj)(c + dj) = (ac + bd) + (bc + ad)j This means ac + bd = 1 and bc + ad = 0. Written in matrix form, this is: \begin{pmatrix} a & b \\ b & a \end{pmatrix} \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} Thus, finding the inverse is equivalent to solving the above system of linear equations. However, the prerequisite is: \det \begin{pmatrix} a & b \\ b & a \end{pmatrix} = a^2 - b^2 \neq 0 Now we have found the meaning of a^2 - b^2: it is the determinant of the coefficient matrix when solving for the inverse!
Note that the above is merely exploration and conjecture, not a proof or a derivation. We see that a matrix appears here. If the determinant of the matrix is positive definite (of course, negative definite would serve the same purpose), then the ring can successfully become a division ring. This matrix is related to the multiplication table of the basis. Generally, each row of this matrix, ignoring signs, is a rearrangement of the numbers in the first row (in the most general case, it should be a linear combination).
Now let us analyze when a determinant can be positive definite. For example, consider a third-order determinant (which corresponds to the existence of a three-dimensional division algebra): \det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei - afh - bdi + bfg + cdh - ceg How can we make it positive definite? A natural idea is to reverse the signs of the negative terms (there are three such terms), which means changing the signs of some numbers in those terms. However, once the signs of these numbers are changed, it will inevitably affect the other three terms, causing the originally positive terms to become negative. This is an irreconcilable contradiction. Therefore, a three-dimensional division algebra over the reals cannot exist. It is worth noting that Hamilton struggled for ten years before realizing this...
However, in a fourth-order determinant, this is possible; otherwise, quaternions would not exist. In determinants of odd order, contradictions similar to the third-order case always arise. Furthermore, if the dimension has an odd factor, similar contradictions occur. Consequently, finite-dimensional division algebras over the real field can only have dimensions of 2^n.
If this article were written before 1860, it might have provided some guidance to friends studying algebra. Of course, 1860 is long gone, and:
In 1861, Weierstrass proved that finite-dimensional division algebras over the real or complex fields, if they satisfy the commutative law of multiplication, can only be the algebras of real and complex numbers (published in 1884).
In 1870, Dedekind reached the same conclusion (published in 1888).
In 1878, Frobenius (F. G. Frobenius, 1849–1917) proved that finite-dimensional associative division algebras over the real field are only the algebras of real numbers, complex numbers, and real quaternions.
In 1881, Peirce also independently obtained the proof. In 1958, using methods from algebraic topology, it was proven that finite-dimensional division algebras over the real field, even including non-associative division algebras, only exist in the four known dimensions: 1, 2, 4, and 8. It can be seen that the real and complex fields possess unique properties.
References
https://en.wikipedia.org/wiki/Normed_division_algebra
http://baike.baidu.com/view/10903855.htm
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