With the uncertain political landscape, and North Korea's continued development of nuclear weapons, this is another interesting Quora question.
The US has already fought a war against North Korea before during the red scare under in an attempt to contain communism - the aptly named Korean war, and hadn’t it been for China sending in a ton of soldiers as the US military neared the Chinese border, the US would have easily won the war.
Granted this was before North Korea’s effort to develop nuclear weapons, and under a different ruler.
The most interesting aspect of a theoretical new war between the US and North Korea is whether North Korea will use nuclear weapons, and if they have the technology to attack US mainland with these. North Korea is not a part of the Non-Proliferation of Nuclear Weapons treaty.
How a war between North Korea and USA would look like strongly depends on three major factors: If North Korea can amass any allies; perhaps most probably China in the event of a US invasion, if North Korea uses nuclear weapons, and Donald Trump.
Although China and North Korea generally has been seemed to be on friendly terms, after North Korea began its nuclear testing program in 2006 China took an opposing stand towards North Korea, and voted for sanctions against North Korea at the UN.
Russia too has voted for further sanctions against North Korea at the UN in response to the escalation of their nuclear program.
With this in mind, it seems unlikely for China, or Russia to ally with North Korea in the event of an attack against US; especially if they send a nuclear attack which they have to do in order to have a chance at winning.
For reference, North Korea spends 25% of its GDP on military of $10 billion. Compare this to the US which in 2015 spent $3.97 trillion on its military; that’s nearly 400 times more than North Korea.
Based on military spending, it’d seem that US would easily defeat North Korea in a traditional war.
But what if North Korea launched a missile towards the US armed with a nuclear warhead (note that there’s currently little evidence of North Korea having the technology required to do such thing)?
Then, the US would have to have to make a decision whether to retaliate using their nuclear arsenal which may lead to a more frequent use of nuclear weapons; not only by the the US, but also other countries as Russia, and China.
On the other hand, if America doesn’t retaliate with nuclear weapons, they risk looking weak and indecisive. The politics involved is way more complicated than I can pretend to know about.
Finally, there’s Donald Trump.
Trump isn’t exactly known to be the most predictably rational person, and if North Korea harrasses him enough, it’s not entirely impossible that he may choose to invade North Korea.
Assuming North Korea isn’t able to gather any allies, America will likely eventually win the war (whatever that means) leaving behind a political landscape that’s difficult to predict, but feasibly may include worsened foreign relations, and a weakened national image.