The book A History of Mathematics: An Introduction
by Victor J. Katz says:
"...probably the most famous mathematical technique coming
from China is the technique long known as the Chinese
remainder theorem. This result was so named after a
description of some congruence problems appeared in one
of the first reports in the West on Chinese mathematics,
articles by Alexander Wylie published in 1852 in the
North China Herald, which were soon translated into both
German and French and republished in European journals..."
(page 222)
This seems to suggest that the name "Chinese Remainder Theorem"
was introduced soon after Wylie's 1852 article.
But the book Historical Perspectives on East Asian Science,
Technology, and Medicine, edited by Alan Kam-leung Chan,
Gregory K. Clancey and Hui-Chieh Loy says:
"A. Wylie introduced the solution of Sun Zi's remainder
problem (i.e. "Wu Bu Zhi Shu") and Da-Yan Shu to the
West in 1852, and L. Matthiessen pointed out the identity
of Qin Jiushao's solution with the rule given by C. F. Gauss
in his Disquisitiones Arithmeticae in 1874. Since then it
has been called the Chinese Remainder Theorem in Western
books on the history of mathematics."
This is ambiguous, as it is not clear whether the author
means that the name "Chinese Remainder Theorem" came into
use after 1852 or after 1874. But the phrasing does suggest
that the name came into use before 1929.
In 1881, Matthiessen published the following article:
L. Matthiessen. "Le problème des restes dans l'ouvrage
chinois Swang-King de Sum-Tzi et dans l'ouvrage Ta Sen
Lei Schu de Yihhing." Comptes rendus de l'Académie de
Paris, 92 :291-294, 1881.
But does the name "Chinese Remainder Theorem"
("le théorème chinois des restes") appear in this article?
Let $L/K$ be an arbitrary extension of fields, and put $G = \operatorname{Aut}(L/K)$.
For a subgroup $H$ of $G$, put $L^H = \{x \in L \ | \ \forall \sigma \in H, \ \sigma(x) = x\}$.
We say that a subgroup $H$ of $G$ is closed if $H = \operatorname{Aut}(L/L^H)$.
Theorem (Artin-Kaplansky): Every finite subgroup $H$ of $G$ is closed.
Moreover, a closed subgroup $H$ of $G$ is normal iff for all $\sigma \in G$, $\sigma L^H = L^H$. If $L/K$ is algebraic, this says that $L^H/K$ is normal.
Also, if $[L:K]$ is finite, then $|G| \leq [L:K]$. So for finite extensions, every subgroup is closed.
If $[L:K]$ is infinite, it need not be the case that every subgroup $H$ of $G$ is closed. In fact, if $L/K$ is algebraic and Galois of infinite degree, then there are always nonclosed subgroups, since any closed subgroup $H$ of $G$ is profinite, hence finite or uncountably infinite. But the group $G$ itself is uncountably infinite, so has countably infinite subgroups.
To see a particular example of a nonclosed subgroup which is moreover normal, take $K = \mathbb{F}_p$ and $L = \overline{\mathbb{F}_p}$, an algebraic closure. Then $\operatorname{Aut}(L/K) \cong \widehat{\mathbb{Z}}$, and the closed subgroups are those of the form $n \widehat{\mathbb{Z}}$: they are all open, and in particular uncountably infinite. Then the subgroup of $G$ generated (not topologically generated!) by the Frobenius automorphism $\operatorname{Fr}: x \mapsto x^p$ is isomorphic to $\mathbb{Z}$, and is a nonclosed normal subgroup whose closure is $\widehat{\mathbb{Z}}$.
Best Answer
The cached page
http://webcache.googleusercontent.com/search?q=cache:q5q43iNq1SQJ:siba2.unile.it/ese/issues/1/690/Notematv27n1p5.ps+normal+basis+theorem&cd=3&hl=en&ct=clnk&gl=us&client=safari
gives some information: Eisenstein conjectured it in 1850 for extensions of finite fields and Hensel gave a proof for finite fields in 1888. Dedekind used such bases in number fields in his work on the discriminant in 1880, but he had no general proof. (See the quote by Dedekind on the bottom of page 51 of Curtis's "Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer".) In 1932 Noether gave a proof for some infinite fields while Deuring gave a uniform proof for all fields (also in 1932).
In Narkiewicz's "Elementary and Analytic Theory of Algebraic Numbers" (3rd ed.) he writes on the bottom of p. 186 that the normal basis theorem is due to Noether, but as usual the history is slightly more complicated.