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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?


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.

Eisenstein integer lattice source

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    $\begingroup$ Jeff Miller's website catalogs this sort of thing:"G.H. Hardy and E.M. Wright refer to the ‘complex’ or ‘Gaussian’ integers"... (An Introduction to the Theory of Numbers, 1938)". There is nothing on Eisenstein integers' naming. $\endgroup$
    – Conifold
    Dec 1 '20 at 12:52
  • $\begingroup$ @Conifold that's quite a resource, thank you! $\endgroup$
    – uhoh
    Dec 1 '20 at 13:22
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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.

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  • $\begingroup$ Thanks for your answer! This is helpful for me generally as well helps to bracket the time frame. $\endgroup$
    – uhoh
    Dec 1 '20 at 12:41

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