How to correspond with academics as a young researcher in high school?
Researchers tend to be very busy advising their own students, and often don't have much free time to commit to supervising others. Recognize that you are asking someone who is very busy to spend some of their limited time on something that (1) may or may not be interesting to them, (2) is not part of their "official" professional responsibilities, and (3) is likely to take time away from their "official" professional responsibilities. (That is why "talk to your advisor" is a common response; they aren't brushing you off, so much as they are redirecting you to someone who is personally responsible for helping you, and has committed to investing that time in you.)

For this reason, emails to researchers asking questions that they don't feel personally responsible for answering, often go unanswered.

You also wrote that you believe the answers to your questions are not going to be immediately obvious, even to an expert:


  I have frequent contact with a professor at my local university, who has taught a class on this subject but has not published any papers or done any research in this area. He and I worked on a few of my questions and were able to solve them, but he couldn't figure out any of my other ones. Although that doesn't preclude "common knowledge to any expert", I am pretty sure it isn't.


which suggests that you are asking for a fairly substantial time investment.

And you are asking for an open-ended commitment in your email:


  I was wondering if perhaps we could start a short correspondence to help me understand what I'm writing about, and to gain more knowledge of class field theory.


I would be extremely reluctant to say "Yes" to this without any clue of how advanced you are in the topic, what you are expecting from me, and whether it would be an interesting or useful correspondence for me.

It would, however, be appropriate to send a short, actionable email, on a question that is directly related to the researcher's published papers. For example:


  Hello Dr. *****,
  
  My name is *****. I'm 14, and I live in South Carolina. I am working on a research project related to Hilbert class fields of global function fields. 
  
  I was wondering if you could answer a question about your paper, [name of paper]. [Specific question that you are looking for an answer to.]
  
  With gratitude, ****
  
  P.S. Would it be OK if I asked some further follow-up questions on this subject?


See 


When asking research-based questions, what are some good practices to maximize the rates at which people reply to emails?
Increase chance of reply when e-mailing an eminent researcher? 


for additional suggestions

For questions that are not related to a specific paper, but are more general questions about the field or your own work (that you and your advisor are unable to answer), I suggest Mathematics Stack Exchange.  People are more likely to invest time in answering a stranger's questions for no professional benefit on a platform where that's what they do.