How did Gaussian and Eisenstein integers get their names?

Question

I can separate this into two questions at some point if necessary, but it's possible that sources for the answer to one will provide the answer to the other at the same time.

I learned about Eisenstein integers after studying this answer to a mathematics problem I'd asked about. Briefly they are represented by a hexagonal lattice on the complex plane, the distance of the six closest points to the origin are all unit length from it. With integers $a$ and $b$ they are

$$a + bu$$

where¹

$$u = \frac{1+ i \sqrt{3}}{2}.$$

Then I learned about Gaussian integers which are represented by a square lattice of length one on the complex plane. With integers $a$ and $b$ they are of the form

$$a + bi.$$

Question: Eisenstein integers are named after Gotthold Eisenstein and I assume Gaussian integers are named after Carl Friedrich Gauss, but who gave these names to these number sets in the complex plane?

Or at least how did consensuses for their names arise?

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¹The linked answer uses that expression for $u$ because that's how the question was formulated. In wikipedia (and likely elsewhere) it is the oblique form (120°) rather than the acute (60°):

$$u = \frac{-1 + i \sqrt{3}}{2}.$$

To make a hexagonal lattice one can use any two of the three unit vectors.

source

Answer 1

In the 19th century, the Gaussian integers were just called "complex integers". In Dirichlet-Dedekind's Vorlesungen uber Zahlentheorie from the late 1800s, the XI-th supplement was the 1st published account about general number fields and its first long section 159 is on elementary number theory in $\mathbf Z[i]$, but this ring was given no special notation (the field $\mathbf Q(i)$ was written as $J$) and had no name like "Gaussian integers".

In Hardy & Wright's An Introduction to the Theory of Numbers, Section 12.2 is called "The rational integers, Gaussian integers, and the integers of $k(\rho)$". Here $\rho$ is a nontrivial cube root of unity, so their unnamed set of integers is the Eisenstein integers. In the notes at the end of that chapter, they write that the Gaussian integers were introduced by Gauss in his work on biquadratic reciprocity and "the numbers $a+b\rho$ were introduced by Eisenstein and Jacobi in their work on cubic reciprocity". It may seem odd that they give the Gaussian integers a special name but don't do the same for what we call the Eisenstein integers, but consider that most of these quadratic rings of integers have no special name: $\mathbf Z[\sqrt{2}]$, $\mathbf Z[\sqrt{-5}]$, and so on. You can't expect each of these to get a special name; at some point it has to stop.

On MathSciNet, the earliest review that mentions Eisenstein integers is a paper by Coxeter and Todd in 1953 and the next two such reviews are for papers from 1969. The fourth earliest reference is a review of the 1970 textbook The theory of numbers. An introduction by Anthony Gioia where the review mentions " "the Jacobian (Eisenstein) integers..." and this is the only review on MathSciNet using the label "Jacobian integers". There are no reviews at all mentioning Eisenstein integers between 1980 and 1997.

Answer 2

The article you linked to gives some historical background: It's whilst Gauss was investigating reciprocity laws that he discovered the Eisenstein and Gaussian integers. The former are the natural domain to study cubic reciprocity and the latter for quartic. He also notes that the integers in higher extensions would help prove higher reciprocity laws.

I don't know who gave them their names but it would be later than 1832 when Gauss introduces both types of numbers in his second monograph on quartic, that is biquadratic, reciprocity.