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In some sense, $\mathbb{C}$ is more than just a field, since aside from the usual field operations, it is also equipped with a conjugation operator $\mathbb{C} \rightarrow \mathbb{C}$.

This means a couple of things.

  1. We can formulate the notion of an inner product space $V$ over $\mathbb{C}$, which includes the axiom

$$\langle x,y\rangle = \overline{\langle y,x\rangle}$$

  1. If $U$ and $V$ are vector spaces over $\mathbb{C},$ and if $f : U \rightarrow V$ is a function, then not only can we make sense of the question "Is $f$ linear?" but also, we can make sense of the question: "Is $f$ conjugate-linear?"

So, I think the notion of "field equipped with a conjugation operator" is very important for linear algebra.

Question. Is there a structuralist/axiomatic definition of this concept? And what about the more general concept of a ring equipped with a conjugation operator? (so that may consider modules over such a ring, and conjugate-homomorphisms between them.)

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    $\begingroup$ Have you heard of Galois theory? $\endgroup$
    – anon
    Nov 5, 2013 at 3:59
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    $\begingroup$ @anon, heard of it, yes. What is the connection? (Pun intended.) $\endgroup$ Nov 5, 2013 at 4:02

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One can do things rather straightforwardly. A fancier statement might be to talk about a field along with an automorphism of the field which is an involution.

This generalizes easily to rings, and in fact the resulting theory is a variety of universal algebra, so the basics are rather well-understood.

This even has a name: a *-ring

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  • $\begingroup$ Is there a special name for an inner product space over a $*$-field (or even a $*$-ring)? $\endgroup$ Nov 5, 2013 at 9:58
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The conjugation map on $\Bbb C$ is a certain type of symmetry. On the one hand, it preserves all elements of $\Bbb R$, and on the other, any equation involving the four basic arithemtic operations remains true if you replace all of the numbers therein with their conjugates. Conjugation is effectively the unique (nontrivial) symmetry of the equational theory of the complex numbers.

Galois theory captures the general situation as follows. Say $F$ is some field and $E/F$ is a field extension (that is, $E$ is a field and $F\subseteq E$). Then we can define the symmetry group ${\rm Aut}_F(E)$, which is to say the group of $F$-algebra automorphisms of $E$. Let's unpackage this information: this means that an element of ${\rm Aut}_FE$ is a function $E\to E$ which preserves the truth of all equations made from the basic operations $\{+,-,\times,/\}$ (division being a partial operation). In particular the function has the "homomorphism property" $f(a+b)=f(a)+f(b)$ and $f(ab)=f(a)f(b)$ for all field elements $a,b\in E$. Furthermore, $f(x)=x$ for all $x\in F$.

Under certain conditions we say $E$ is Galois over $F$. When this happens, the subgroups of the so-called Galois group $G_{E/F}:={\rm Aut}_FE$ are in lattice correspondence with intermediate fields which lie between $E$ and $F$. The correspondence is between pointwise stabilizers and fixed fields.

In particular, $G_{\Bbb C/\Bbb R}=\Bbb Z/2\Bbb Z$. The trivial element is the identity map and the nontrivial element is complex conjugation. In this way, Galois theory vastly generalizes the concept of conjugation: indeed given an algebraic number, the elements of its orbit under a Galois action are called its conjugates. If you have ever "rationalized the denominator" by "multiplying by conjugates," you were implicitly employing basic Galois theory! This defines a norm for number fields too: multiply a number by all of its conjugates.

Galois groups can be much richer and more complicated than $\Bbb Z/2\Bbb Z$. An open problem states that every finite group is a Galois group of a number field, in fact. Thus the "field operators" that are present here need not be involutions, there can be more than one, they need not commute, etc.

I wouldn't say that complex conjugation is a big thing in linear algebra, as much as it is a ubiquitous item in many contexts where linear algebra is also of the essence. IIRC some initial treatments of linear algebra stick mostly to real numbers in fact, and motivate complex numbers via algebraic closedness (and hence eigenvalues and various matrix decompositions).

Rather than Galois theory being useful for linear algebra, it is the other way around. And Galois theory in turn is useful for algebraic number theory. The complex numbers inhabit a very special place in mathematics and analysis in particular. That is why we see $\Bbb C$ and complex structures (like Riemann surfaces) in differential geometry but not arbitrary number fields. (On the other hand some of the deepest levels of number theory have broached algebraic geometry, which is very geometric in spirit. Neukirch famously claimed number theory is geometry.)

Even more generally, rings automatically come equipped with automorphism groups, and their linearized parents, endomorphism rings. In abstract algebra and beyond you can pick and choose all sorts of rings and symmetries and actions ("operators") along these lines.

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