Round 9: Tossup 11

An ordering of terms within these mathematical objects is denoted by the shorthand “grevlex.” A set of these objects is iteratively transformed by Büchberger’s algorithm to obtain a Gröbner basis. Solution sets to systems of equations defined by these objects are called algebraic varieties. Hilbert’s (10[1])basis theorem applies to rings of these objects, (10[1])which are formed by adjoining dummy (10[1])variables to rings like the complex numbers. Gauss’s lemma concerns the irreducibility of these objects, whose solvability can be determined via their Galois (“gal-WAH”) groups. (10[1])Vieta’s formulas provide explicit relations (10[1])between solutions to these objects, which have no generic solution in degree five or higher (10[1])by the Abel–Ruffini theorem. For 10 points, the fundamental theorem of algebra concerns the roots of what functions? ■END■

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