Prime Minister Menachem Begin's brazen decision to "legalize" three illegal Jewish settlements on the Arab West Bank, at potentially exorbitant political cost to President Carter's Mideast peace hopes, has widened a crack in the administration's once-solid Mideast front.

That crack, still concealed under a blanket of uniformity imposed by the White House, opened as a result of Carter's deference during Begin's state visit. Whatever the President's long-range diplomatic objective in letting Begin seize and hold a remarkable initiative over him during the Israeli leader's visit here, it triggered a backfire throughout the administration - everywhere except the President's own Oval Office.

Begin's rapid decision to exploit Carter's good will by "legalizing" those three settlements ("an absolutely unacceptable move," one top-level Carter advisor told us) led to an official State Department rebuke - but only a mild presidential demurrer in which Carter unaccountably blamed himself.

Pressed at his press conference to explain why Begin had so quickly seen fit to set back the President's peace efforts, Carter threw a protective arm around Begin and said, "I did not think about talking to him" on the specific question of "legalizing" the three settlements.

Yet high officials who participated in the talks between the President and prime minister told us Carter and Secretary of State Cyrus Vance "absolutely" covered all contingencies involving the incendiary settlements question even if the word "legalization" had not been used. "There was no chance of misunderstanding on Begin's part," one official told us. The United States wanted no "legalizing" of existing settlements.

The response of the President, so much softer than the cold anger privately expressed elsewhere in his government, may now lead to the establishment of new settlements (not just legalization of old ones) in a continuation of what may be called a "policy of preemption" by the new Israeli government.

Adding weight to this warning is the fact that Begin long ago placed Gen. Erik Sharon, hero of the Yom Kippur war but a political primitive regarded by some Israeli leaders as unstable, in overall charge of the government's settlements policy.

The radical religious parties (on which Begin's slim majority in the Knesset rests) are demanding new settlements. Sharon can point to Carter's gentle wrist-slap of last week to argue his case that Israel will not be taking much risk with Carter in setting up new settlements (at least 12 are blueprinted for instant occuption).

It may be, instead, that some Mideast diplomats here are correct in arguing that Begin needed a sop for the religious radicals, considering his campaign promise to create settlements all over the Israeli-occupied West Bank of the Jordan. Now that the sop has been offered, that should put an end to it until the possibilities of a resumed Geneva peace conference have been fully explored.

But other diplomats argue that Begin's personal political history points the other way. Although not a political gambler, Begin is a shrewd oddsmaker of unusual courage and conviction. He knows, as one of his staunchest supporters here told us, "there's nothing much left in Washington of the original Carter Middle East policy, hardly a trace of Jimmy's old demands for withdrawals to the 1967 borders, or the Palestinian homeland."

Begin, the wartime terrorist shrewd and brave enough to calculate correct odds on blowing up Jerusalem's King David Hotel may now calculate that Carter can be pushed a good deal further on the settlements question. As the President said last week, new settlements "provide obstacles for peace," but they are "obstacles which I think we can overcome" - scarcely a warning of retribution for establishment of new settlements.

Other aspects of Carter's evolving Middle East policies have caused confusion here. For example, he confided to Italian Prime Minister Giulio Andreotti last week that he fully agreed with the June 29 Common Market statement calling for a Palestinian homeland.

That surprised experts who knew Carter had worked overtime to head off that Common Market resolution and, failing that, to tone it far down.

The spilt inside the administration over proper handling of Israel's flouting of the United States on the settlements question may only be beginning. Indeed, as Vance flies to the Middle East, some politicians here wonder whether he can count any longer on the White House as a home base to back him up.