O ct 2 00 5 HILBERT 90 FOR BIQUADRATIC EXTENSIONS
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چکیده
Here something stubborn comes, Dislodging the earth crumbs And making crusty rubble. It comes up bending double, And looks like a green staple. It could be seedling maple, Or artichoke, or bean. That remains to be seen. —Richard Wilbur, " Seed Leaves " (1–8) Whether reading or writing mathematics, it is over the simplest theorems that we linger. Something quickens at the hope that such a theorem, a result of already uncommon brevity, might grow into a result of uncommon grandeur. Might it become still more wondrous, maturing into a figure as elegant as it is encompassing? Hilbert's Theorem 90 is just such a theorem. In its simplest form, in the case of a quadratic extension of fields, Hilbert's Theorem 90 is a seedling, and turning it over in our hands, we naturally speculate about the mature form we might one day behold. Most commonly, friends tell us to expect from the seed what is affectionately known as " Hilbert 90 " : a result in the theory of cyclic extensions. Those who have surveyed Galois cohomology, however, know that A. Speiser sensed something different, a cohomological result in a non-cohomological era. Recognizing the power of this result, E. Noether put it to use, and the result became known as Noether's Theorem. More recently, others have imagined the mature form being
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1 1 O ct 2 00 5 HILBERT 90 FOR BIQUADRATIC EXTENSIONS
Here something stubborn comes, Dislodging the earth crumbs And making crusty rubble. It comes up bending double, And looks like a green staple. It could be seedling maple, Or artichoke, or bean. That remains to be seen. —Richard Wilbur, " Seed Leaves " (1–8) Whether reading or writing mathematics, it is over the simplest theorems that we linger. Something quickens at the hope that such a theore...
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