From the Artin's book, it says that we can "construct" the complex number by the quotient ring $\frac{R[x]}{<x^2+1>}$, and we can prove that there is an isomorphism between complex number and this quotient ring.

My question is: how can we say two things are equivalent by just using an isomorphism?

Here are some thinkings:

  1. $x$ and $e^x$ are also isomorphic with respect to "$\times$" of $x$ and "$+$" of $e^x$, but $x$ and $e^x$ are totally different two functions. We can see that their graphs are different, and we can not say that we "construct" $e^x$ by finding an isomorphism.

  2. What about other properties of sets? like the topological property, two sets are isomorphic doesn't mean that they are homeomorphic. (I'm new to topology and please fix me if I'm wrong)

  3. Assume we don't use the isomorphism to "construct" complex number and we just define the elements in $\frac{R[x]}{<x^2+1>}$ as a complex number, which means that every complex number is an equivalence class. And since complex numbers contain real numbers, it also means that real numbers are equivalence class, which is not beautiful, and we don't know that if the axiom of completeness still holds on a set of equivalence class.

  4. We can also use the direct definition of complex numbers like $i^2=-1$, but this definition doesn't reveal the essential of complex numbers. I think it should be related to polynomials and is a structure of polynomials because the complex numbers first showed up when we solve the cubic equation and we can't avoid it to get all roots.

  5. I know that we get new fields just using the isomorphism, and I understand that we just need the algebraic structure for computation because isomorphism is convenient and helpful for computation. But things are different when we try to construct such a basic math structure like complex number, it is used everywhere so I think we need to preserve every property between two sets, which is the equivalence between sets.

And since all the properties may not be preserved between set, I think we can not say we construct complex numbers, we can only say their algebraic structures are the same.

I've also checked this link Difference between equality and isomorphism to see what's the difference between isomorphism and equivalence but it is not related to what I want to ask.

This is the first time I ask question here, please let me know that if there is any problem in my question. Any thinking and answer will be appreciated. Thank you.

  • $\begingroup$ "the complex numbers first showed up when we solve the cubic equation" - no, they are needed already to solve quadratic equations. $\endgroup$ – Paul Frost Aug 13 '20 at 22:22
  • $\begingroup$ @PaulFrost, it depends on what 'needed' means. Historically, people were perfectly willing to say that the quadratic $\lambda^2 + 1 = 0$ simply had no solutions $\lambda$; but $\lambda^3 - 15\lambda - 4 = 0$ visibly has $\lambda = 4$ as a solution, and yet applying Cardano's formula to it involves complex numbers. $\endgroup$ – LSpice Apr 12 at 16:20

Reply to 1, 2:

A notion of isomorphism is relative to what structure you are preserving. $\mathbb N$ is isomorphic to $\mathbb Z$ in terms of cardinality, but not in terms of, say, additive structure [$\mathbb N$ does not have additive inverses, $\mathbb Z$ does have additive inverses].

So we start with some axiomatization of complex numbesr [say, the field which is the algebraic closure of the reals] We then show that the definition $\mathbb R[X]/(x^2 + 1)$ is isomorphic to the above mentioned structure.

Reply to 3:

Regarding having "extra things" inside this definition of $\mathbb C$ as an equivalence class: Indeed. That's the nature of set theory. We also have that $0 \in 5$ if we use the Von Neumann construction of naturals. For a lengthy debate on this, see 'Set theory without junk theorems' on MathOverflow.

Reply to 4

The "direct definition" makes no sense. What is $i$ and how does one set it to $-1$? The rigorous way of doing this, of course, is to construct $\mathbb R[x]/(x^2 + 1)$: (i) we add a "free element" called $x$, (ii) we force $(x^2 + 1) = 0$ by quotienting. Thus we have effectively added an element $x$ such that $x^2 = -1$.

Reply to 5

Why do we need to preserve equivalence between structure of sets? We never "peek into" the set theoretic structure of the complex numbers; We use their axiomatic properties


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