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I am interested in the history related to algebraic numbers and have two questions:

  1. Who first proved that algebraic numbers form a field?
  2. Who first proved that algebraic numbers form an algebraically closed field?

Let us recall that a complex number $z$ is algebraic if $z$ is root of a non-constant polynomial with integer coefficients.

Wikipedia only lists important mathematicians that wrote influencial books on Algebraic Number Theory: Gauss, Dirichlet, Dedekind, Hilbert. So, I could suggest that someone between Gauss and Dedekind already knew that algebraic numbers form an algebraically closed field. But is there any more precise historic information?

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    $\begingroup$ Part of the problem in answering a question like this is that mathematicians like Gauss did not use the terminology "field" or thought in those terms. For example, Galois "showed" that if one looks at all powers of a primitive root of an irreducible polynomial (over $\mathbb{Z}$, but that's always what "a polynomial" meant to him) one gets a "field". But he did not call it this at all; for him it was just saying that one could multiply and add these numbers "as usual". The usual theorems about "Galois fields" comes more than half a century after Galois, by E. H. Moore. [contd.] $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Mar 31, 2023 at 9:56
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    $\begingroup$ [contd.] But I don't doubt that the first statement 1), if stated in terms familiar to him (e.g. the product of two algebraic numbers is algebraic, etc. etc.), would have been known to Gauss. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Mar 31, 2023 at 9:58
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    $\begingroup$ On a related subject, you might ask "who first proved the fundamental theorem of Galois theory?" (in the sense of lattice anti-isomorphisms). The answer is van der Waerden in 1930. Galois (a century earlier) did not prove any correspondence between subgroups and intermediate field extensions. But with time it is not so hard to derive it from what he did prove. Thus attributing it to Galois is not entirely wrong. I suspect a similar pattern will appear in answers also to this question. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Mar 31, 2023 at 10:05
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    $\begingroup$ @Carl-FredrikNybergBrodda Thank you for the comments. So, I can attribute this result to Gauss (in a popular lecture) and what about the second question? That roots of polynomials with algebraic coefficients are algebraic? Was this fact also familiar to Gauss? And are there any evidences of this in his books of papers? $\endgroup$
    – Taras Banakh
    Commented Mar 31, 2023 at 10:44
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    $\begingroup$ @Carl-FredrikNybergBrodda you're saying van der Waerden in 1930 was the first to prove the fundamental theorem of Galois theory in the sense of lattice anti-isomorphisms because it appeared first in his book Modern Algebra. But van der Waerden's book was based on lectures he attended of Artin and Noether, and it is really Artin who first formulated the Galois correspondence in terms of that lattice anti-isomorphism. See Kiernan's The Development of Galois Theory from Lagrange to Artin (Arch. Hist. Exact Sci. 30 (1971), 40-154). $\endgroup$
    – KConrad
    Commented Apr 2, 2023 at 23:00

1 Answer 1

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In 1882, Dedekind and Weber developed the theory of Riemann surfaces purely algebraically, by taking as their primary object of study not Riemann surfaces, but instead function fields $K$ over $\mathbf C$. The points of the Riemann surface (smooth projective algebraic curve over $\mathbf C$) corresponding to $K$ were defined to be what we'd describe in modern language as the discrete valuations on $K$ that are trivial on $\mathbf C$, and Dedekind and Weber developed everything they needed using ideas inspired by algebraic number theory. (Their paper has been translated into English by Stillwell here).

Dedekind and Weber wrote in the introduction to their work that everything they had done would remain true if the role of complex numbers is replaced everywhere by the algebraic numbers. That reflects the fact that classical algebraic geometry can be developed over algebraically closed fields (at first, of characteristic $0$), so it seems fair to say that by the time of this work of Dedekind and Weber, some people understood that the algebraic numbers form an algebraically closed field.

Dedekind and Kronecker each had a concept of field. Dedekind used the term Körper and had in mind by this subfields of the complex numbers. Kronecker used fields of complex numbers and fields of rational function in several variables, which he put together under the label "rationality domain" (Rationalität-Bereiche). Kronecker had a constructivist mindset, not believing in $\pi$ the way we would, so he would not have considered a field -- even inside the complex numbers -- as defined conceptually as a set with some closure properties. Dedekind freely worked with infinite sets defined by some properties (like Dedekind cuts of rational numbers to define real numbers or ideals in a number field).

Look in treatments of algebraic number theory by Dirichlet, Dedekind, and Kronecker to see if they show that every root of a (monic) polynomial with algebraic coefficients is algebraic. That amounts to understanding, in an older language, that the algebraic numbers are algebraically closed.

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    $\begingroup$ Thank you for mentioning Stillwell's translation of Dedekind and Weber's paper. I wasn't aware of it, and now am salivating at the thought of reading it. $\endgroup$
    – pinaki
    Commented Apr 3, 2023 at 3:06
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    $\begingroup$ Thank you for the answer from which I understand that the most probably Gauss was the first who realized that algebraically closed numbers form a field (more precisely he knew that the sum and product of algebraic numbers is algebraic), but the algebraic closedness of the field of algebraic numbers should be credited to Dirichlet (1805-1859), Dedekind (1831-1916) or Kronecker (1823-1891). Dedekind was the youngest of them and he was a student of Gauss and a younger colleague of Dirichlet editing the works of Dirichlet, so he knew what was done by Gauss and Dirichlet very well. $\endgroup$
    – Taras Banakh
    Commented Apr 3, 2023 at 7:04
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    $\begingroup$ I do think that Kronecker's image of $\pi$ was very much like ours. Where we differ is the image of the real numbers as an infinite set. $\endgroup$
    – Franz Lemmermeyer
    Commented Apr 6, 2023 at 16:26

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