Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and van der Waerden.

MathOverflow Asked by Squid with Black Bean Sauce on November 9, 2021

3 AnswersFor a pre-20th century textbook: Modern Higher Algebra by George Salmon (1876) has exercises (with solutions).

If I may broaden the query from "abstract algebra" to more general "algebra", I note that Elements of Algebra by Euler (1770) has more than one hundred exercises. The exercises are discussed in The origin of the problems in Euler’s algebra.

Here is an example, from the chapter on cubic equations [source].

And another example (find an integer $x$ such that $2x^2-5$ is a cube) [source]

Answered by Carlo Beenakker on November 9, 2021

Two important early 20th-century abstract algebra textbooks that were superseded by van der Waerden are:

Hasse, H., 1926. *Höhre Algebra*.

Haupt, O., 1929, *Einführung In Die Algebra, Zweiter Band* - Mit Einem Anhang Von W. Krull, Akademische Verlagsgesellschaft M. B. H., Leipzig.

For an in-depth historical account of the transition from algebra to abstract algebra (including discussions of the various textbooks) see:

Leo Corry, 1996, *Modern Algebra and the Rise of Mathematical Structures*, Birkhäuser Verlag.

Answered by Philip Ehrlich on November 9, 2021

I think the following references might be useful:

H. Weber (1895/1896): Lehrbuch der Algebra, 2 volumes. Vieweg, Braunschweig. It includes examples, but it does not have any separate exercises.

E. Artin (1938): Foundations of Galois Theory. New York University Lecture Notes, New York.

N. Bourbaki (1947): Éléments de Mathématique, Algèbre. Hermann, Paris.

I am not totally sure whether the last two references include exercises.

Answered by MaryS. on November 9, 2021

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