This ad for courses in Korean says that Korean is an official language in South Korea (and North Korea inter alia), and the English language edition of Wikipedia also states that, but doesn't have any citations for that. It also mentions the National Institute of Korean Language, which doesn't have any citations, apart from a link to the institute.
By contrast, Politifact states that South Korea (and North Korea inter alia) don't have an official language.
Is Korean an official language in South Korea?
Note that an official language is different from a national language.