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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 work with infinite sets having properties (like Dedekind cuts of rational numbers to define real numbers).

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.

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.

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 be remain true if the role of complex numbers is replaced everywhere by the algebraic numbers. So certainly 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 work with infinite sets having properties (like Dedekind cuts of rational numbers to define real numbers).

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.

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 work with infinite sets having properties (like Dedekind cuts of rational numbers to define real numbers).

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.

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 be remain true if the role of complex numbers is replaced everywhere by the algebraic numbers. So certainly 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).

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.

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 be remain true if the role of complex numbers is replaced everywhere by the algebraic numbers. So certainly 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 work with infinite sets having properties (like Dedekind cuts of rational numbers to define real numbers).

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.