State education is by definition socialized education. From Jacob G. Hornberger at fff.org:
It would be virtually impossible to find a better example of socialism here in the United States than the public schooling systems that exist in every U.S. state. Ironically, it is this socialist system that is primarily responsible for the widespread belief among non-libertarians that “the United States has never been a socialist country,” as New York Times columnist Timothy Egan stated in a recent NYT op-ed. (See my two recent articles “A Life of the Lie on Socialism” and “Socialism in America, 31 Years Ago.”)
It is worth noting that public schooling is a core feature of the educational systems in Cuba, North Korea, and Vietnam, all three of which are widely known as socialist countries. That’s because public schooling is a socialist system.
Perhaps it’s also worth noting that while we call it “public” schooling, a more accurate name for it is government schooling or state schooling. That’s because state and local governments own and operate the educational systems. If state and local governments owned and operated churches, would we call them public churches or state churches?
Under public schooling, the government owns, operates, controls, and dictates the provision of education in society. In a purely socialist system, like North Korea, this means that every child in the nation is required to receive his education in a government facility.