What are polynomial like expressions that have complex numbers for the exponents? Got curious about polynomials and Galois theory the other day and realized I have no idea how current mathematics treats polynomials (or rather polynomial like expressions) that have arbitrary algebras for the exponents. A quick search yields polynomial extensions like Laurent polynomials but I couldn't find anything that uses any other groups for the exponent, like Gaussian integers, complex numbers, cyclotomic rings, hypercomplex numbers and other algebras.
What are these structures called and how is their behavior similar and different with ordinary polynomials?
 A: These are basically commutative group rings (or more generally monoid rings), or limits thereof.
Let $G$ be a group and $R$ a ring. Intuitively, $R[G]$ is the $R$-algebra of all finite $R$-linear combinations of elements of $G$, with the most obvious definition addition and multiplication:
$$\left(\sum_{g\in G}a_gg\right)+\left(\sum_{g\in G} b_gg\right)=\sum_{g\in G}(a_g+b_g)g$$
$$\left(\sum_{g\in G}a_gg\right)\left(\sum_{h\in G}b_hh\right)=\sum_{g,h\in G}a_gb_hgh=\sum_{g\in G}\left(\sum_{uv=g}a_ub_v\right)g.$$
Equivalently, $R[G]$ is the set of functions $G\to R$ with finite support, made into an $R$-algebra using pointwise addition and convolution as multiplication.
One can expand the set of elements allowed via $R[[G]]:=\varprojlim R[G/N]$. Note this is not the same as the profinite completion's group ring, $R[\,\varprojlim G/N]$, it is much more expansive. Here we take a system of 'compatible' functions $G/N\to R$ ranging over $N\le G$ with $[G:N]<\infty$ and endow the set of these systems with pointwise addition and convolution algebra structure.
If $G$ is abelian and written additively then we can put the actual elements into exponents of some formal variable, so elements look like $\sum a_gT^g$ with $T^gT^h=T^{g+h}$. Things are a lot nicer in an abelian setting: in point of fact, using representation theory (Artin-Wedderburn) and abelian group theory, if $G$ is finite abelian then the complex group algebra is isomorphic to a ring direct product of $|G|$ copies of $\bf C$; more specifically,
$${\bf C}[G]=\bigoplus_{\chi\in\widehat{G}}\left(\frac{1}{\sqrt{|G|}}\sum_{g\in G}\chi(g)g\right){\bf C}\cong{\bf C}^{|G|}$$
is an explicit isomorphism of $\bf C$-algebras ($u{\bf C}$ is the $\bf C$-linear span of $u\in {\bf C}[G]$ and $\widehat{G}$ the dual).
Going from finite to finitely generated doesn't change much; one obtains ${\bf C}[T_1^{\pm1},\cdots,T_r^{\pm1}]^{\oplus|t(G)|}$ where $r$ is the rank of the free part and $t(G)$ the torsion subgroup of $G$ (which will be finite).
If you want the take the group ring of the underlying additive group of a ring (like the Gaussian integers, as you say), the above theory no longer applies. It is nice that the structure is abelian, but the lack of being finitely-generated (for the exponent group) changes things. Rational exponents do occur in nature, albeit with restrictions pertaining to the support of the underlying functions $\rm exponents\to scalars$ being represented by the polynomials (more generally, series).
For example the field of Puiseux series is the algebraic closure of the field of Laurent series. Over at the secret blogging seminar I read that ${\bf C}_p$ (the metric completion of the algebraic closure of ${\bf Q}_p$, the $p$-adic metric completion of $\bf Q$, which is the $p$-adic analogue of $\bf C$ and is in fact abstractly isomorphic to it i.e. not topologically but field-theoretically) is not "spherically complete," and Poonen's thesis showed that the spherical completion $\Omega_p$ can be represented as the set of all power series in $p$ with Teichmuller digits (i.e. roots of unity for coefficients) whose exponents are well-ordered subsets of the rationals.
Anyway, back to reality, the isomorphism class of $R[T^S]$ as a ring is determined by $R$ and the isomorphism class of $S$ as a $\bf Z$-module (as the multiplicative structure of $S$ makes no appearance); in particular if $S\cong \bigoplus A_i$ is a finite direct sum as groups then $R[T^S]\cong\bigotimes_R R[T^{A_{\large i}}]$, and more generally if $S\cong\bigoplus A_i$ then $R[T^S]=R[T^{A_{\large i}}]_i$ (a formal variable $T_i$ is adjoined for every $i$ with exponents from $A_i$, possibly infinitely many $i$). This sort-of reduces the classification of these sorts of polynomial rings to the classification of indecomposable modules. Considering ways of expanding elements to include formal power series (such that the algebraic operations are well-defined; one can't square $\sum_{n\in \bf Z}T^n$ for example) makes things even more complicated.
As a starting point for structure theory, one may one to consider $R[T^S]$ for indecomposable groups $S$ and prime rings or fields $R$ (i.e. $\bf Z$, $\bf Q$, ${\bf F}_p$). Depending on the arithmetic properties of $S$ (like its order and exponent) and the characteristic of $R$ one may observe inseparability phenomena, which is known to be hard to characterize in general in field theory alone. Verify or falsify usual ring properties: integrality, algebraic closedness, PID/UFD, dimension, Noetherianness or Bezoutness or Dedekindness, etc. This seems to be the type of information most directly relatable to your question, but I am unfortunately not familiar with literature that covers these sorts of tasks.
A: If $R$ is any ring and $G$ is any group, then the group ring $R[G]$ is ring whose elements are finite formal sums of terms labelled by elements $g\in G$ and with coefficients in$~R$. If the group $G$ is written additively, then one can write such terms as $rX^g$ where $X$ is a formal indeterminate, since multiplication of terms is given by $rX^g*sX^h=rsX^{g+h}$; this resembles polynomials with coefficients in $R$ and exponents in$~G$. If however $G$ is already written multiplicatively, it is better to write terms simply as $r.g$. The construction is also valid for a monoid in the role of$~G$ and is then called a monoid ring; the case of the additive monoid$~\Bbb N$ gives the usual polynomials over$~R$.
The main difference with usual polynomials, or with Laurent polynomials, is that there is no such thing as evaluating a group ring element $\sum_i r_iX^{g_i}$ at some value $X=a\in R$. The reason is that there is not in general any way to define exponentiation $a^g$ in a way consistent with the multiplication of formal terms, namely such that $a^ga^h=a^{g+h}$ for all possible values $g,h\in G$, when $G$ goes beyond the set of natural numbers or (in case $a$ is invertible) the set of integers. Problems already appear in such simple cases as $G=\Bbb Q$ (the additive group), even with "nice" coefficient fields $R=\Bbb R$ or $R=\Bbb C$: there is no consistent way to define rational powers for negative reals or of general complex numbers so that the usual laws of exponentiation are valid in all cases. In fact one can for $R=\Bbb C$ salvage just the rule $z^az^b=z^{a+b}$ (even with $a,b\in\Bbb C$) that does not require any relation between evaluations at different values, by choosing a branch of the complex logarithm and putting $z^a=\exp(a\ln z)$ for this choice; extending this linearly does define a ring morphism $\Bbb C[\Bbb Q]\to\Bbb C$ of "evaluation of polynomials with rational exponents at a fixed number$~z$". Even so, the corresponding "polynomial function" of$~z$ cannot be made to be continuous on$~\Bbb C$, and there will also be failure of other algebraic identities that one may expect, such as $z^{-a}=(z^{-1})^a$ for $z\neq0$ (taking $z=-1$ and $a=\frac12$ shows the impossibility to have this) or $(xy)^a=x^ay^a$. You can find many questions on this site related to this fundamental impossibility to define exponentiation for not-positive-real base and non-integer exponent in a way consistent with the usual rules manipulating powers (and roots, viewed as fractional powers).
Of course going beyond $G=\Bbb Q$ things only get more problematic. Since evaluation is one of the most important things we do with polynomials, this makes working in group or monoid rings rather different than working with polynomials, and for this reason one does not call monoid ring elements polynomials although they are formally defined in much the same way. Note however that $R[\Bbb N^k]\cong R[X_1,\ldots,X_k]$, so in this case the definition amounts to that of (multivariate) polynomials, for which evaluation (in $k$-tuples of values) is well defined.
