Bonus
If n is 2, these numbers are 1 and -1. For 10 points each:
[10m] Name these numbers that, when raised to the power of a positive integer n, equal one. These numbers are sometimes named after the French mathematician Abraham de Moivre (“mwahv”).
ANSWER: roots of unity [accept nth roots of unity]
[10e] The sum of the n-th roots of unity for any n always equals this number. Any number multiplied by this number yields this number.
ANSWER: zero [or 0]
[10h] Adjoining a root of unity to the rationals is the definition of a cyclotomic one of these constructs, which is a ring where every non-zero element has a multiplicative inverse.
ANSWER: fields [accept cyclotomic fields; accept cyclotomic field extension; prompt on cyclotomic extension by asking “extension of what?”]
<UCSD A, Other Science>
| Editions | Heard | PPB | Easy % | Medium % | Hard % |
|---|---|---|---|---|---|
| 2 | 103 | 15.53 | 87% | 37% | 31% |
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| Claremont A | UC Irvine | 10 | 10 | 10 | 30 | MEH |
| Claremont B | USC B | 10 | 0 | 10 | 20 | MH |
| UC Santa Barbara | UCLA D | 0 | 10 | 0 | 10 | E |
| UCLA B | UCLA A | 10 | 10 | 0 | 20 | ME |
| UCLA C | UCLA E | 10 | 10 | 10 | 30 | MEH |
| UCLA F | USC A | 0 | 0 | 0 | 0 | |
| UCSD B | Claremont C | 10 | 10 | 0 | 20 | ME |
Summary
| Tournament | Edition | Exact Match? | Heard | PPB | Easy % | Medium % | Hard % |
|---|---|---|---|---|---|---|---|
| UK (North) | UK | Y | 5 | 18.00 | 80% | 60% | 40% |
| UK (South) | UK | Y | 8 | 15.00 | 88% | 38% | 25% |
| Northern California | US | Y | 4 | 25.00 | 100% | 100% | 50% |
| Southern California | US | Y | 7 | 18.57 | 71% | 71% | 43% |
| Eastern Canada (1) | US | Y | 4 | 17.50 | 100% | 25% | 50% |
| Florida | US | Y | 4 | 15.00 | 100% | 25% | 25% |
| Great Lakes | US | Y | 10 | 15.00 | 80% | 20% | 50% |
| Lower Mid-Atlantic | US | Y | 9 | 13.33 | 100% | 22% | 11% |
| Upper Mid-Atlantic | US | Y | 11 | 14.55 | 91% | 27% | 27% |
| Upper Mid-Atlantic | US | Y | 1 | 30.00 | 100% | 100% | 100% |
| Midwest | US | Y | 9 | 13.33 | 67% | 44% | 22% |
| North | US | Y | 4 | 10.00 | 75% | 0% | 25% |
| Northeast | US | Y | 2 | 10.00 | 50% | 50% | 0% |
| Pacific | US | Y | 8 | 13.75 | 88% | 13% | 38% |
| Southeast | US | Y | 12 | 15.83 | 100% | 42% | 17% |
| Upstate NY | US | Y | 5 | 18.00 | 100% | 40% | 40% |