THE GREAT RESOURCE RUSH; Alaska

Among the letters that recently piled up on Gov. Jay S. Hammond's desk was a note from Barbara Carey, a county commissioner from Florida.

Mindful of the Alaska's recent oil riches, she offered a suggestion on how the state could help out its distant Southern friend and boost its own tourism drive: Alaska could bankroll the new sports complex her Dade County was planning to build in Florida. It would become an ''architectural advertisement'' for the 49th state.

''Governor,'' she wrote, ''how would you like the idea of all of these national telecasts opening with the phrase, ''. . . and now, from the beautiful Alaska Sports Complex in Miami, we bring you . . . .''

Mr. Hammond, a former bush pilot, trapper, and part-time poet, replied with an ''Owed'' (as in ode) to Barbara Carey that included a few penny-wise phrases of its own:

While it's true that we briefly were riding high

on a ballooning oil wealth bubble,

total dependence on price of oil

can lead to a barrel of trouble!

For example, it takes just one . . . A-rab

who's swallowed a rancid date

or fallen off of his camel's back

to clobber this forth-ninth state.

The pen pal exchange sums up a major challenge facing Alaska today: The euphoria of oil wealth can melt like an iceberg in August.

In fact, any illusions Alaskans may have had of unlimited energy fortunes are bumping into the harsh realities of tumbling world oil prices, migrating job-seekers, and social strains from unequal growth.

Enough petrodollars have flowed into the treasury from state-owned Prudhoe Bay in the past decade to start Alaska on its way to underwriting the most dramatic changes in its brief but colorful history.

But now the state is being forced into a more miserly mood. The crimp in world oil prices has eroded the earning power of its Arctic bank vault--source of some 85 percent of state funds.

Though far from putting Alaska into the poorhouse, the decline in affluence is sharpening the focus on larger questions the state faces in managing its roller-coasting wealth:

* How best to smooth out the seasonal and historic bumps in the economy. They began with the fur-trade boom of the 1700s, were followed by the Klondike gold rush a century later, and have peaked in today's oil age.

* How to use the oil bounty to improve the lot of average Alaskans. Residents of some rural areas just recently caught their first glimpses of television in their own villages.

* How to spur development of otheb mineral and energy resources without creating ''state subsidized'' industries and without marring some of the world's most hauntingly beautiful real estate.

To some officials, including Mr. Hammond, the current anxiety may bring some much-needed setting of priorities. But, adds the salt-and-pepper-bearded governor, ''I'd just as soon have the point driven home with a carpenter's hammer instead of with an 18-pound mall.''

Alaska today stands as something of an impoverished billionaire. On the one hand, Prudhoe Bay has become a honey pot producing more money than even the most glittery-eyed Klondikers could have ever imagined.

Oil rigs on the tundra-shawled North Slope pump enough crude to dump close to current revenue projections tumbling monthly, the state is expected to take in $ 53 billion from Prudhoe between now and 1998. (That is down $24 billion from a projection made in January.)

A dip in future fortunes may leave some highways shorter than expected or the balance in the state's ''savings account'' a little lower than some might like. But after Prudhoe Bay, there may still be pools of crude to tap. Oil companies are now gearing up for a new era of exploration. As much as half the country's undiscovered oil is believed to lie in the area, although almost no one expects another bonanza like the current one.

On the other hand, the hardships are many in this unwieldy state. Thinly settled with 400,000 of the country's most doggedly independent people, Alaska is an endless expanse of Arctic desert, glacially clawed mountains, and misty fjords. All this makes for spectacular scenery but taproot-tough economic conditions.

Bigger than Texas, California, and Montana combined, the state has fewer roads than Rhode Island. Health care in many rural villages consists of yearly visits by a public nurse. Many communities still lack running water.

Costs in Alaska are as high as moose antlers, but so are wages. Toothpaste in Bethel, a wind-swept community in the treeless tundra of western Alaska, where goods have to be brought in by plane or barge, goes for $2.85 a tube. A bag of potato chips is $3.25, gasoline $1.63 a gallon.

''People on the outside look at us and say, 'Look at those people up there, they're blowing $4 billion,' '' says Ed Dankworth, a burly former state trooper, who is now chairman of the state's Senate Finance Committee. With Mozart for background music, he explains, ''We started without toilets. We started without sidewalks.''

Some oil wealth is being funneled to the people. In recent years, the state has wiped out income taxes and handed local governments enough largess to cut property and sales taxes. It has set up programs offering low-interest loans to homebuyers as well as providing cheap money to college students, businesses, and farmers.

Along with the cheap loans and low taxes come the new buildings, many of them in Anchorage, home of half the state's population. A far cry from the tent city it was 65 years ago, today it has as many mirror-skinned buildings as any other city its size. Residents are awaiting a new performing-arts center and talking about opera. Plenty of Alaskans criticize Anchorage as a patch of misplaced urban sprawl in their untamed state. But a frontier flavor persists, with an occasional moose loping into town.

Enough oil money has been put into blacktop and other amenities to help perk up the economy. Last year statewide employment grew 4.5 percent, producing a larger work force than during peak construction of the Trans-Alaskan pipeline. Job growth is expected to continue this year. But there is an Achilles' heel: Whenever word spreads of job openings, fortune-chasers flock in from around the country.

Before you pack to head north, consider this: The state's official unemployment rate still tops 10 percent. Many who migrated here to work on the Alaska natural gas pipeline--still at least two years away--are spending their time in Fairbanks playing euchre.

Beyond that, there are plenty of Alaskans who don't feel they are getting their fair share of the oil wealth at all. Some rural residents grouse that too much money is flowing to oil company headquarters in Texas, or going into ''monuments'' in Anchorage. ''We just don't feel the impact out here,'' says Ivan M. Ivan, a Yupik Eskimo and president of an association of 56 western communities.

Thunderclaps of protest also arise from many of those concerned about the state's pristine environment. For many conservationists, concern about land use and lack of planning has replaced the battle over dividing up the state's lands. Their worry: Breakneck spending will dangerously boost the population and hasten the decline of the environment as well as the state's ''frontier'' way of life.

''With all the wealth, it's like kids with a whole bunch of candy,'' complains Cindy Marquette, director of the Alaska Center for the Environment.

Moreover, the oil age is accelerating cultural changes among Alaska's natives (Aleuts, Eskimos, and Indians), who are seeing snowmobiles and Sony Walkmans show up in villages used to gill-net fishing and whale hunting.

Yet if the current drop in oil revenues is giving Alaskans more time to think through the pace and scope of their ''blueprint for progress,'' it could also, potentially, help smooth troubled relations with Washington. Waist-deep in recession, some areas of the ''Lower 48'' would like to see Alaska shoulder more of its own financial burden, not to mention some of theirs. The dip in revenues gives the state political ammunition in warding off such attacks.

In the long run, the state has a treasure-trove of energy, mineral, and other resources from which to build a bridge to tomorrow--free from the swings of world oil prices.

Alaska's opportunities, in short, remain virtually unlimited--but the risks are many. ''It's a lot trickier to handle more wealth than you need at the moment because you can get into a lot more trouble,'' says Governor Hammond.

Translated, Barbara Carey of Dade County, Fla., that means there will be no ''Alaska Sports Complex.''