roots of unity - 2025 ACF Fall

Question

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?”]

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Summary


Northern California	Main	Y	4	25.00	100%	100%	50%
Southern California	Main	Y	7	18.57	71%	71%	43%
Eastern Canada (1)	Main	Y	4	17.50	100%	25%	50%
Florida	Main	Y	4	15.00	100%	25%	25%
Great Lakes	Main	Y	10	15.00	80%	20%	50%
Lower Mid-Atlantic	Main	Y	9	13.33	100%	22%	11%
Upper Mid-Atlantic	Main	Y	11	14.55	91%	27%	27%
Upper Mid-Atlantic	Main	Y	1	30.00	100%	100%	100%
Midwest	Main	Y	9	13.33	67%	44%	22%
North	Main	Y	4	10.00	75%	0%	25%
Northeast	Main	Y	2	10.00	50%	50%	0%
Pacific	Main	Y	8	13.75	88%	13%	38%
South Central	Main	Y	6	13.33	100%	17%	17%
Southeast	Main	Y	12	15.83	100%	42%	17%
UK (North)	UK	Y	5	18.00	80%	60%	40%
UK (South)	UK	Y	8	15.00	88%	38%	25%
Upstate NY	Main	Y	5	18.00	100%	40%	40%

Data


Berkeley Skeleton King	Berkeley Little Prince	10	10	0	20
Berkeley Boss Bandit	Stanford A	10	10	0	20
Stanford B	UCSC	10	10	10	30
Berkeley Mighty Miner	Berkeley Archer Queen	10	10	10	30
Claremont A	UC Irvine	10	10	10	30
Claremont B	USC B	10	0	10	20
UCLA B	UCLA A	10	10	0	20
UC Santa Barbara	UCLA D	0	10	0	10
UCLA C	UCLA E	10	10	10	30
UCLA F	USC A	0	0	0	0
UCSD B	Claremont C	10	10	0	20
McGill B	McGill C	10	10	10	30
Ottawa D	McGill A	0	10	10	20
Carleton A	Ottawa C	0	10	0	10
McGill E	Ottawa A	0	10	0	10
Florida A	Florida B	10	10	10	30
South Florida A	Florida State A	0	10	0	10
Florida Tech A	Florida Tech B	0	10	0	10
UCF A	ERAU Daytona A	0	10	0	10
Pittsburgh	Michigan A	0	10	0	10
CWRU D	Ohio State D	0	10	10	20
Louisville B	WVU A	0	10	10	20
CWRU B	Louisville C	0	10	10	20
CWRU C	Michigan C	10	10	10	30
Michigan D	WVU B	0	10	0	10
Louisville A	Michigan State B	0	0	0	0
CWRU A	Ohio State A	0	0	0	0
Ohio State C	Michigan State D	0	10	0	10
Michigan B	Kentucky	10	10	10	30
William & Mary B	Liberty B	0	10	0	10
JMU B	Liberty A	0	10	0	10
UNC C	NCSU B	0	10	0	10
Virginia Tech B	Tusculum A	0	10	0	10
VCU A	UNC A	10	10	0	20
Virginia Tech C	Virginia A	0	10	0	10
Virginia Tech A	Duke A	10	10	10	30
William & Mary A	William & Mary C	0	10	0	10
UNC D	Liberty C	0	10	0	10
George Mason B	George Mason C	0	10	0	10
Penn	Maryland B	0	10	0	10
Maryland C	Columbia A	0	10	0	10
NYU A	Rutgers C	10	0	0	10
NYU B	Maryland A	0	10	10	20
Rutgers A	NYU C	10	10	0	20
George Mason A	Rowan	0	10	0	10
Lehigh A	Rutgers D	0	10	0	10
Stony Brook	Princeton	10	10	10	30
Vassar A	Lehigh B	0	10	0	10
Rutgers B	Vassar B	0	10	10	20
Johns Hopkins	Columbia B	10	10	10	30
Illinois Orange	WashU Grizzlers	0	10	0	10
Purdue A	Chicago A	10	10	10	30
Illinois Canada	William Woods University A	0	10	0	10
Iowa B	Rose-Hulman A	0	0	0	0
Illinois Blue	Purdue B	10	0	10	20
Purdue C	Illinois Urbana	10	10	0	20
Indiana A	Purdue D	0	10	0	10
Illinois Champaign	WashU Grolars	10	0	0	10
Iowa A	Indiana B	0	10	0	10
Iowa State	Wisconsin B	0	10	0	10
Macalester	Minnesota	0	10	0	10
Wisconsin A	St. Thomas	0	0	10	10
Michigan Tech	Wisconsin C	0	10	0	10
Southern Connecticut State University A	Boston University B	0	0	0	0
Alberta A	Texas	0	10	10	20
Boise State	Alberta B	0	0	0	0
Arizona State A	Washington Masu	0	10	10	20
Arizona State B	Washington Chinook	0	10	10	20
British Columbia B	Washington Coho	0	10	0	10
British Columbia C	Washington Sockeye	10	10	0	20
Washington Chum	Washington State	0	10	0	10
Washington Atlantic	Washington Steelhead	0	10	0	10
Williams A	Tufts A	10	10	0	20
Tulsa A	Nebraska B	10	10	10	30
Central Oklahoma A	Kansas State A	0	10	0	10
Murray State College B	Harding	0	10	0	10
Oklahoma	Central Oklahoma B	0	10	0	10
Rose State	Nebraska C	0	10	0	10
Tulsa B	Murray State College A	0	10	0	10
Vanderbilt A	Auburn B	0	10	0	10
Clemson A	Auburn A	10	10	0	20
Georgia Tech B	Alabama A	10	10	10	30
Georgia Tech C	Georgia Tech F	10	10	0	20
Mississippi State A	Chipola College	0	10	0	10
Tennessee Tech	Sewanne	0	10	0	10
South Carolina A	Georgia A	0	10	0	10
Tennessee A	Emory A	0	10	0	10
Georgia Tech D	Tennessee B	10	10	10	30
Vanderbilt B	Clemson B	0	10	0	10
Georgia Tech E	Vanderbilt C	0	10	0	10
Georgia Tech A	Auburn C	10	10	0	20
Durham D	Durham B	10	10	10	30
Sheffield A	Durham C	0	10	0	10
Sheffield B	Edinburgh B	0	10	0	10
Edinburgh A	York	10	0	0	10
Edinburgh C	Edinburgh D	10	10	10	30
Imperial D	Birmingham	0	10	0	10
Bristol	Imperial C	10	0	0	10
Cambridge C	Cambridge A	0	10	0	10
LSE	Cambridge D	0	10	0	10
Cambridge E	Southampton	10	10	0	20
Imperial A	Cardiff	0	10	10	20
Imperial B	Oxford	0	10	0	10
Cambridge B	Warwick	10	10	10	30
Cornell D	Binghamton C	0	10	0	10
Binghamton B	Binghamton A	0	10	0	10
RIT A	RIT B	10	10	10	30
Rochester	Cornell C	10	10	10	30
Cornell B	ESF	0	10	0	10