The nation's unemployment rate dropped to 10.4 percent in January, its first decline since the recession began a year and a half ago and one of several clear signs of improvement in the job market, the government reported yesterday.

For the first time, the Labor Department yesterday also published jobless figures that included the military. Since no military personnel are out of work, their inclusion had the effect of lowering the January jobless rate to 10.2 percent.

President Reagan and other administration officials hailed the figures as evidence that the economy is "on the mend."

Labor Department analysts cautioned, however, that the drop in unemployment between December and January may have been exaggerated by special, seasonal factors.

An estimated 11.4 million Americans were out of work last month, after seasonal adjustment, the Labor Department said. The December jobless rate was 10.8 percent, with more than 12 million people counted as unemployed. If the military had been included in December, the rate would have been 10.7 percent.

Black unemployment remained at its postwar peak of 20.8 percent in January, the report said.

Yesterday's employment news supports the growing evidence that the recession is ending, or is already over. Reagan's chief economist, Martin Feldstein, said yesterday that the 10.8-percent December jobless figure likely was "the peak unemployment rate of this recession."

However, Feldstein and other analysts warned that the rate "may bounce back up" from the January level.

The AFL-CIO said yesterday that there was no real improvement in unemployment last month because the decline was caused primarily by people dropping out of the labor force, rather than finding jobs.

Janet L. Norwood, commissioner of the Bureau of Labor Statistics, said, "I think the seasonal adjustment is reflecting the real world, but it has exaggerated" the improvement between December and January. "The labor market has improved, but . . . not as much as these data suggest," she told the Joint Economic Committee of Congress.

Unusually mild January weather boosted construction activity above seasonally normal levels, while reduced levels of Christmas sales meant less hiring than usual in December, and thus fewer firings of temporary workers in January. These seasonal factors could be reversed in coming months.

Moreover, there are still 1.8 million "discouraged" workers who would like to find jobs but have stopped looking.

Last month also saw a big rise, from 5.4 million to 6.8 million, in the number of people forced to work only part time.

President Reagan's official forecast for the economy predicts that the jobless rate will average 10.7 percent this year--on the new basis of calculation that includes the military--and will decline to 10.4 percent by the fourth quarter.

The president has been accused of producing an overly pessimistic economic forecast in order to pressure Congress into approving more spending cuts.

The latest economic data suggest that Reagan's forecast for economic growth of 3.1 percent during 1983 may prove too low. Feldstein, chairman of the Council of Economic Advisers, said last week that the economy may grow by as much as 5 percent during 1983, bringing unemployment down to about 9 1/2 percent by year-end.

One puzzling feature of yesterday's jobless report was a sharp, 579,000 drop last month in the size of the labor force, which almost matched the 590,000 decline in those reported as unemployed.

The jobless rate for adult men fell from 10.1 percent to 9.6 percent, but instead of a rise in the total number at work, there was a slight drop.

Good news in yesterday's employment release included a 340,000 increase in payroll employment outside of farming. There were also marked increases in the average length of the workweek in the nation's factories, and in the private sector as a whole.

The factory workweek rose from 38.9 to 39.7 hours, "the first real improvement in this leading indicator since early 1981," Norwood said.

Most of last month's employment increase was accounted for by increased jobs in construction and trade. However, "the highly cyclical manufacturing industry," which has borne much of the brunt of recession, "leveled off in January and small job increases occurred in many of the individual manufacturing industries," Norwood said.

Hourly earnings, adjusted for shifts in overtime and between high- and low-paying industries, climbed 0.4 percent, the Labor report said. Average weekly earnings rose by 1.5 percent in the month.

Unemployment rates for blacks, adult women (9.0 percent) and Hispanics (15.5 percent) were little changed during the month.

The jobless rates for manufacturing workers fell by 1.8 percent to 13 percent, and for construction workers from 22 percent to 20 percent, after seasonal adjustment.