The announcement came as a top United Nations human rights official told a Security Council meeting that it was "essential" that Pyongyang be referred to the International Criminal Court.
But outside observers were skeptical, saying that such an advance in nuclear technology seemed unlikely.
Analysts in recent years have believed that North Korea may have been working toward -- but didn't yet have the capability to produce -- a hydrogen, or thermonuclear, bomb. It can be hundreds of times more powerful than an atomic bomb.
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North Korean leader Kim Jong Un made the claim, according to state media outlet KCNA, while touring a historic weapons industry site in the reclusive communist country.
Will China protect North Korea?
North Korea had become "a powerful nuclear weapons state ready to detonate [a] self-reliant A-bomb and H-bomb to reliably defend its sovereignty and the dignity of the nation," Kim said, according to the KCNA report.
The claim is "inextricably linked" to the Security Council meeting, said Jasper Kim, director of the Center for Conflict Management at Seoul's Ewha University.
"North Korea wants to achieve a stronger bargaining position."
