View source for Noether's theorems/History/Noether's position and the writing of the paper

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== Her position at Göttingen ==

Emmy Noether held no post at Göttingen in 1918. She had come at the invitation
of Hilbert and Klein in 1915, and the faculty had declined to let her habilitate.
Her lectures were announced under Hilbert's name.

She was an algebraist. The work that made her reputation, and that mathematicians
mean when they say her name, is in ring theory and ideal theory and comes later.
The 1918 paper sits outside that line.

It does not sit alone, though. She published a second paper in the Göttingen
''Nachrichten'' in the same year, ''Invarianten beliebiger
Differentialausdrücke'', at pages 37 to 44, presented by Klein at the session of
25 January 1918 — the same session at which Klein presented his first note. That
paper is on the invariants of arbitrary differential expressions, and it is the
bridge between her algebraic work and the July paper. Any account that treats
''Invariante Variationsprobleme'' as an isolated excursion has overlooked the
January paper.

== The writing ==

Noether spent part of the spring of 1918 in Erlangen, visiting her widowed and
ailing father. Because she was away, her correspondence with Klein survives, and
it records the work in progress.

A postcard of 15 February already sketches what became the second theorem,
though only in a particular case. A letter of 12 March gives a preliminary
formulation of the essential consequence of it, for a variational problem
invariant under a subgroup of an infinite-dimensional group. By then the shape
of the result was fixed.

On 23 July she presented her results herself, to the Göttingen Mathematical
Society. On 26 July Klein presented the paper to the Göttingen Scientific
Society, which was the more consequential venue because its audience was not
restricted to pure mathematicians. The printed paper carries the line recording
this, naming Klein and the session of 26 July.

The dedication is to Klein, on the fiftieth anniversary of his doctorate.

== A caution about the July sessions ==

Neither published text is the text that was presented.

Noether's paper carries a footnote stating that the final version of the
manuscript was not submitted until the end of September. Klein's second note
carries a footnote saying that his manuscript did not reach final form until the
middle of September.

Both papers were therefore revised for two months after their presentations, in
a period when each author was reading the other. What was said at either session
is not recoverable from the printed papers, and this volume makes no claim about
it.

== Klein's account of her theorems ==

When Klein reprinted his second note in his collected works, he added a footnote
that states both of Noether's theorems and gives her formulation of Hilbert's
assertion about energy.

The footnote is worth its own attention. It restates the first theorem: that
invariance of an integral under a group with a finite number of essential
parameters makes that many linearly independent combinations of the Lagrange
expressions into divergences. It restates the second: that invariance under a
group depending on arbitrary functions, entering to some order of derivative,
produces that many identical relations among the Lagrange expressions and their
derivatives to that order. And it gives the precise form of Hilbert's claim — the
energy relations become improper if and only if the integral is invariant under
an infinite group containing the translation group as a subgroup.

This matters for provenance. The most consequential result in the paper is
attested twice from primary text: in Noether's own final section, and in Klein's
footnote to his own paper. Two authors, two documents, one statement. Very
little in the history of this subject is that well anchored.

''Provenance: the German original of Noether's paper was read in full and is
public domain. Her prose is paraphrased throughout and marked as paraphrased; no
English wording of hers is transcribed anywhere in this volume. The
correspondence dates and the account of the Erlangen visit are taken from the
secondary literature, not from the letters. Klein's paper was read in a modern
English translation whose terms of reuse have not been established, and is
summarised, not quoted. See [[Noether's theorems/Sources|Sources]].''