As North Korea races to build a weapon that for the first time could threaten American cities, its neighbors are debating whether they need their own nuclear arsenals. The North’s rapidly advancing capabilities have scrambled military calculations across the region, and doubts are growing the United States will be able to keep the atomic genie in the bottle. For the first time in recent memory, there is a daily argument raging in both South Korea and Japan — sometimes in public, more often in private — about the nuclear option, driven by worry that the United States might hesitate to defend the countries if doing so might provoke a missile launched from the North at Los Angeles or Washington. In South Korea, polls show 60 percent of the population favors building nuclear weapons. And nearly 70 percent want the United States to reintroduce tactical nuclear weapons for battlefield use, which were withdrawn a quarter-century ago. There is very little public support for nuclear arms in Japan, the only nation ever to suffer a nuclear attack, but many experts believe that could reverse quickly if North and South Korea both had arsenals. Prime Minister Shinzo Abe has campaigned for a military buildup against the threat from the North, and Japan sits on a stockpile of nuclear material that could power an arsenal of 6,000 weapons. Last Sunday, he won a commanding majority in parliamentary elections, fueling his hopes of revising the nation’s pacifist Constitution. This brutal calculus over how to respond to North Korea is taking place in a region where several nations have the material, the technology, the expertise and the money to produce nuclear weapons. Beyond South Korea and Japan, there is already talk in Australia, Myanmar, Taiwan and Vietnam about whether it makes sense to remain nuclear-free if others arm themselves — heightening fears that North Korea could set off a chain reaction in which one nation after another feels threatened and builds the bomb. In a recent interview, Henry A. Kissinger, one of the few nuclear strategists from the early days of the Cold War still living, said he had little doubt where things were headed. “If they continue to have nuclear weapons,” he said of North Korea, “nuclear weapons must spread in the rest of Asia.” “It cannot be that North Korea is the only Korean country in the world that has nuclear weapons, without the South Koreans trying to match it. Nor can it be that Japan will sit there,” he added. “So therefore we’re talking about nuclear proliferation.” Such fears have been raised before, in Asia and elsewhere, without materializing, and the global consensus against the spread of nuclear weapons is arguably stronger than ever. But North Korea is testing America’s nuclear umbrella — its commitment to defend its allies with nuclear weapons if necessary — in a way no nation has in decades. Similar fears of abandonment in the face of the Soviet Union’s growing arsenal helped lead Britain and France to go nuclear in the 1950s. President Trump, who leaves Nov. 3 for a visit to Asia, has intensified these insecurities in the region. During his presidential campaign, he spoke openly of letting Japan and South Korea build nuclear arms even as he argued they should pay more to support the American military bases there. “There is going to be a point at which we just can’t do this anymore,” he told The New York Times in March 2016. Events, he insisted, were pushing both nations toward their own nuclear arsenals anyway. Mr. Trump has not raised that possibility in public since taking office. But he has rattled the region by engaging in bellicose rhetoric against North Korea and dismissing talks as a “waste of time.” In Seoul and Tokyo, many have already concluded that North Korea will keep its nuclear arsenal, because the cost of stopping it will be too great — and they are weighing their options. Long before North Korea detonated its first nuclear device, several of its neighbors secretly explored going nuclear themselves. Japan briefly considered building a “defensive” nuclear arsenal in the 1960s despite its pacifist Constitution. South Korea twice pursued the bomb in the 1970s and 1980s, and twice backed down under American pressure. Even Taiwan ran a covert nuclear program before the United States shut it down. Today, there is no question that both South Korea and Japan have the material and expertise to build a weapon. All that is stopping them is political sentiment and the risk of international sanctions. Both nations signed the Nuclear Nonproliferation Treaty, but it is unclear how severely other countries would punish two of the world’s largest economies for violating the agreement. South Korea has 24 nuclear reactors and a huge stockpile of spent fuel from which it can extract plutonium — enough for more than 4,300 bombs, according to a 2015 paper by Charles D. Ferguson, president of the Federation of American Scientists. Japan once pledged never to stockpile more nuclear fuel than it can burn off. But it has never completed the necessary recycling and has 10 tons of plutonium stored domestically and another 37 tons overseas.The New York Times