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These are notes to pp. 1-11 and 21-49 of Artin’s classic—”the last corrected printing of the 1944 second revised edition,” republished by Dover in 1998—to assist the study of his account of Galois theory as far as the Fundamental Theorem. In some places I have followed Artin’s revised German edition (Teubner, 1959), which seems not to have been translated into English. I hope that any mistakes that may be noticed will be brought to my attention.

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This is a version of an elementary lecture I gave recently. At the end some questions raised or implied but not answered in the lecture are posed as exercises; these are cited along the way as Q1, Q2, etc. Some of them are less gentle than the lecture.

If symbols are not legible, refreshing the page usually restores them.

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If they were to invest the seaport of Syracuse, the Athenians had to build a wall that would surround it on the landward side. They were on the point of accomplishing this, when the Spartan Gylippus arrived with his forces just in time to interfere with the work, so effectively that the Syracusans were able to push their counter-wall past the Athenian line. Thus investment became impossible, and the task of the Athenians so much greater that it was to prove beyond their power. A dramatic moment, fate in the balance, “on a razor’s edge,” in Homeric phrase! —perhaps magnified for effect by the historian, in service to human vanity, which plumes itself on decisive action in crisis. παρὰ τοσοῦτον μὲν αἱ Συράκουσαι ἦλθον κινδύνου, comments Thucydides (7.2): So near to danger did Syracuse come, within so little distance of it, his expression fitting the narrowness of wall and time. He had used it once before (3.49), of the port of Mytilene, that one successfully walled in (3.18) and besieged by the Athenians but likewise saved by timely arrival, of word from Athens that the remaining townsfolk should be spared (a countermand anticipating the counter-wall).

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It is by no means only the introduction of arbitrary subdivisions (mere arithmetical adjustment) that removes the “natural” unit of length from nature. No, the Earth itself refuses to sanction the aspirations of the founders of the metric system.

Since the 17th century it has been known that the length of a pendulum beating seconds varies with its location on the Earth; specifically, with the latitude. This is because the constant of proportionality between time and the square root of length depends on the acceleration of gravity g—the rate at which the speed of a freely falling body increases—and g varies with latitude. Hence if the pendulum is to deliver a unit of length, the arbitrary choice of the second of time must be supplemented by something still less justifiable: the selection of a place on the Earth to put the pendulum.

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In this part, “wretched matter” begins to assert itself.

If we want to measure length, we need a unit of length. Given such a unit, not only lengths, but areas and volumes as well, can be expressed in terms of it: foot, square foot, cubic foot. The founders of the new system desired a natural unit, one that would not depend upon an arbitrary choice, and that would be perpetually available in the natural world, so that it could readily be consulted in case of doubt. Neither criterion would be satisfied by (for example) the length of the king’s foot.

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