Fairfax City residents will vote Tuesday in the most bntterly divisive elction since the small city of 21,000 broke away from surrounding Fairfax County in 1961.

With 16 candidates running for the six-member city council, and three for mayor, the ballot also is the longest in the city's history.

The election appears to turn on the school question, an issue that has been simmering now for 17 years and got several thousand residents boiling mad a few months ago when the present mayor, Nathaniel Young, teamed with three council members to put the city on a seperate school course.

The crucial action was the council's refusal to discuss with the county a new school contract aunder which the county would operate city schools. A petition of more than 4,000 city residents urging the council to reopen negotiations was ignored by a majority of teh council.

The issue of whether the city stays with the Fairfax County school system or sets up a seperate school system for its 4,500 students will get a final public airing tonight at the last of several candidates meetings. Sponsored by the parent Teacher Organization, it will be held in Fairfax High School at 7:30 p.m.

A Washington Post poll of city residents last month found a clear majority (55 percent) wants to remain with the county school system. The poll found only 32 percent of city residents approve of a separate city school system and 13 percent were not sure or hadn't followed the issue closely enough to form an opinion.

While the school issue has divided the candidates into two clear groups, they also appear to be divided on issues of mass transit and regionalism. One group appears generally separatist, and even antagonistic, in its relations with Fairfax County and Metro. The other is largely in favor of improving relations with the county and rejoining the Metrobus system (but not subway) the city withdrew from last year.

Two incumbent council members are running for mayor against Young - Walter L. Stephens Jr., who says he agrees with Young on the school question and most other issues, and Frederick W. Silverthorne, who has argued strongly for staying in the county school system.

joining Silverthorne is an informal slate of six council candidates, including incumbent members Susanne W. Max and Lee H. Wigren, all of whom are either members of or endorsed by Citizens for Continued Quality Education.

Formed from parent teacher organization within the city, CCQE has become the rally organization for residents who want the new city council to negiotiate with Fairfax County over schools. Council candidates Carl Hemmer, John Perrin, Glenn White and Bill Scott, as well as Silverthorne, helped form the group.

The group has criticized the mayor and present council majority for putting the city on an "isolationist path" and urges city voters to elect leaders "willing to solve our problems through cooperation with surrounding jurisdictions," according to a CCQE flyer.

CCQE also has criticized as biased and inaccurate the publication Cityscene, the city newsletter written by the city manager's staff and delivered door-to-door to residents.

All of the 10 other council candidates, as well as mayoral candidates Young and Stephens, favor leaving the Fairfax County school system. A parallel organization to CCQE, Save Our City, has sprung up backing Mayor Young and eight of the 10 candidates favoring independent schools.

Signers of SOC pamphlets include, beside Young, incumbent council member Leonard A. Mobley and candidates Will H. Carroll, Ronald E. Escherish, Theodore Grefe, Donna F. Jenkins, Clifford W. Overcash, A. Ramsay Stallman and William F. Willoughby.

Stephens and two council candidates, incumbent Frederick J. McCoy and Dee Ann R. Schmidt, are not officially allied with SOC. Both McCoy and Schmidt say they favor separate schools and are against rejoining the Metrobus system but do not want to be tied to any positions taken by SOC. McCoy's campaign literature describes him as a "peace candidate" who seeks "peace and an end to enmity with our neighbors," and urges voters to support Schmidt and three of the 10 SOC council candidates.

Part of the school debate has been over quality, with CCQE insisting the city should remain with the highly rated-county school system and SOC candidates saying that county schools aren't all that good, each group citing various studies and tests to support its thesis.

A major part of the debate over the school issue has been over money, but the figures are almost all questionable.

It is not clear how much more Fairfax County would insist upon as the city's fair share of school costs. (Last fall when negotiations broke down the county had said the city should pay an additional $1.6 million a year.)

Nor is it clear exactly how much it would cost the city to run an idependent school system of its own. Three brief studies have said at least as much as more than the city ahs been paying the county. But CCQE predicts it might cost the city $3 million more a year and says Falls Church, which has a small city school system of its own, last year spent 44 percent more per pupil on its school children than did Fairfax City under the county school system.

SOC, on the other hand, says the cost of a city-run school system depends upon what citizens want but that whatever they want it would cost less than what the county wants the city to pay.