Can North Korea's nuke poison the planet with radiation?

No. From everything I have read, their nukes are explosive (designed to land and destroy), but not dirty (i.e. not designed to spread radiation wantonly.

Terrorist bombs are assumed to be “dirty”.

It could affect the area. They said to want to test one in the atmosphere. There is a world wide agreement that no nukes be detonated in the atmosphere.

Whichever system delivers the warhead, will be highly unproven. The technology is very new to NK, and very few tests of the bombs/warheads, or missiles were done compared to say the US or Russia.
If they launch it on a missile, it could break apart, blow up on launch, fall on an unintended target etc… The entire project, is extremely dangerous.

Okay…time for a Radiation 101 lesson. In short, the answer is yes, but not badly. Any nuclear bomb testing will release certain radioactive isotopes into the atmosphere. Radioactive Iodines, for example, are frequently seen following nuclear reactions. Cesium-137 and/or Cesium-134 can be by-products of nuclear fission as well.
The daughter product (those isotopes produced by nuclear fission) curve for most US nuclear plants show a high production of Xenons and Kryptons. But US nuclear plants use enriched U-235 which is a thermal fuel. Nuclear bombs generally use either U-235 or Plutonium-239. If Plutonium is used, the daughter products will be slightly different.
Another consideration is where the bombs are detonated. If they are detonated underground, the amount of contaminants they throw into the air are limited. If the bomb is detonated at ground level or in the air (over the Pacific Ocean, for instance), then all the by-products are likewise blasted into the air. Air currents can carry these isotopes quite a ways. When Chernobyl melted down, it released a huge cloud of radioisotopes into the air. The impact of those radioisotopes (how far they traveled) could be seen all over the globe. There isn’t anywhere in this country that you cannot find Cs-137 that is from bomb testing back in the 50’s. Bananas are a great source of potassium, but K-40 shows up in all bananas as well as in people that eat them. K-40 can be found naturally, but only in low levels. Most comes from bomb testing back in the 50’s as well.
But in the long run, Mother Nature has a way of dealing with most of the trash we throw at her. Most radioactive isotopes are relatively short lived. There are the exceptions like I have mentioned…Cs-134, Cs-137, K-40, and tritium (H-3). Another factor to consider is how these radioisotopes might be an impact to humans. Are they getting into some food chain? Are they hazards if inhaled? Do they have strong gamma radiation decays? There are many, many, variables that would make me question whether I should be concerned and what precautions I should take.
Do I want humans testing nuclear weapons? Hell no! Nuclear weapons are one of those things I wish we could go back and un-invent. Am I concerned about the physical impact of one or two nuclear tests? Not really.

No. If they did, the test blasts from the US, China, and USSR would have done that long ago, back in the 50’s and 60’s.

It depends on how close you are and what you consider bad. Some people, such as my university genetics professor, considered all radioactive pollution bad, and said nuclear technology overall was increasing genetic mutations worldwide. Every living thing in close proximity to a blast would have been affected in not-good ways. It’s a matter of perspective.

@kritiper I happen to believe that many of our current issues of today are a direct result of the Atomic Bombs. Just to mention a few, the number of children born with ADHD and Asbergers, the increase in diabetes cases, the increase in the number of people with gender identity issues, and many neurological disorders, including Alzheimer’s and migraines..

^^ sorry, most of the really bad isotopes decay in the magnitude of days, if not hours. The issues you address are likely a hodgepodge of increased identification and awareness, diet (especially processed food), medication and environmental toxin induced disease. Radiation from atomic weapons is known to increase the incidence of cancer but this effect drops off precipitously with time.

@YARNLADY Are you sure all you mention are not caused by just all the water-borne chemicals and birth control drugs, and other pollutants in the world’s water supplies? Maybe some of the problems are caused by atomic test blasts of old, but the world would not be noticeably “poisoned” and one or two more smaller ones. @ARE_you_kidding_me ‘s last post was right on target.

@kritiper Good points. Things just keep on piling up.