The State Department's East-West trade expert resigned yesterday to protest attempts by hawks within the Reagan administration to toughen controls on the sale of oil and gas exploration equipment to the Soviet Union. Hours later, however, he was persuaded to reconsider his abrupt resignation.

William A. Root, described as a prickly, thoroughly professional diplomat who has held the key job of director of the State Department's Office of East-West Trade through a number of administrations, agreed to reconsider his resignation after meeting with Commerce Undersecretary Lionel H. Olmer.

"I hope he stays," said Olmer, in charge of international trade for the Commerce Department.

Olmer added that Root apparently decided to resign as a result of long frustrations over what he saw as attempts by hawks within the administration to block efforts, headed by Secretary of State George P. Shultz, to forge a coordinated policy on East-West trade with Western Europe and Japan.

The administration's unilateral decision in December, 1981 to impose trade sanctions on the sale of natural gas pipeline equipment to the Soviets created a major rift within the Western alliance that ultimately forced the administration to withdraw the ban.

Commerce Assistant Secretary Lawrence J. Brady, a strong proponent of stiff curbs on East-West trade who often disagrees with Olmer, his boss, said Root "failed to recognize the massive change in the attitude on the part of our allies toward export controls.

"Bill Root has been an impediment to this administration's attempts to strengthen the Cocom system (through which the Western allies control the sale of strategic materials to East bloc nations and the Soviet Union) from the very beginning," continued Brady.

"We will strengthen export controls," he declared, "and it will be done in spite of Bill Root."

The Root-Brady differences illustrate the bitter splits within the administration over how much military technology flows to the Soviet war machine through East-West trade as as well as the value of trade curbs as an instrument of foreign policy.

These splits are erupting now in the publicly-voiced dismay from the right wing of the Republican Party, to which Brady belongs, over President Reagan's failure to impose stiff sanctions on the Soviets for the downing of a South Korean passenger plane.

The Soviet downing of a Korean jetliner has strengthened the more hawkish members in the administration, said one Republican congressional staff aide. This new stregnth may manifest itself in the coming debate over extending the Export Administration Act that allows export controls to East Bloc nations. The act expires Sept. 30.

The move that apparently precipitated Root's resignation was a recommendation this week by the administration's Advisory Committee on Export Policy, headed by Brady, that effectively would have given the Defense Department a veto over sales of oil and gas exploration equipment to the Soviet. These controls presently fall under the joint jurisdiction of the State and Commerce Departments.

The recommendation would have imposed the tougher national security controls on oil and gas exploration products instead of the foreign policy controls they presently fall under. National security controls had been reserved for sales of high technology until the Carter administration used them for a Soviet grain embargo to avoid congressional oversite.

William Schneider Jr., undersecretary of state for security assistance, science and technology, said the change had been recommended to strengthen the United States' hand in negotiating with its allies in Cocom, who were being asked to place national security controls on 17 different types of oil and gas technology.

Congressional sources, however, saw this as a move by hawks within the administration to change the president's policies and said the recommendation, if followed, "totally destroys the effort the State Department is making within Cocom, to get multilateral controls."

Brady said no export licenses have been denied as a result of his committee's recommendations, "but I am not saying that what previously would have been approved will not be denied."

Olmer, who takes a more middle ground on East-West trade than Brady, said a subcabinet-level committee recomendation "doesn't mean the policy they recommend will get implemented."

Moreover, changes in the list of products on the foreign policy and national security control may be made only by Commerce Secretary Malcolm Baldrige and Secretary of State Shultz. "No one does it but the big fellows," Olmer said.

Baldrige and Shultz prevailed last month in a dispute that went to the president over the sale of pipelaying equipment to the Soviets. Although Defense Secretary Caspar Weinberger opposed the sale, Reagan removed the licensing requirement for their sale.