From the lead, front-page news article in today's New York Times: "the speculation about firing Mr. Powell, a move that could turn the independent central bank into a political tool, has undermined confidence in a pivotal institution essential to economic policy."
This juxtaposition of "independent" and "political" is a false dichotomy. The Federal Reserve is political in the sense that it was established by a law passed by elected politicians, in Congress, who also passed the Humphrey-Hawkins Act of 1978 instructing the Fed to pursue stable prices and maximum employment. One man's "independent" is another man's "unaccountable."
The notion that it was "the speculation about firing Mr. Powell" that "undermined confidence" in the Fed is questionable, too. If everyone were so confident in the Fed, there wouldn't be any need for speculation about firing the chairman. Maybe it's the Fed's own performance that is undermining confidence and stirring speculation about its chairman's future. It's a bit of a "which came first, the chicken or the egg" sort of situation.
Maybe instead of firing Powell, Trump ought to try to fire Bernanke and Yellen, since they are the ones who got the Fed into the situation that Powell is trying to extricate the Fed from.
Anyway, it's natural that a president whose core appeal was based on the line "you're fired" would wonder whether he can fire the Fed chairman. The question goes back to President Franklin Roosevelt, who tried to fire a Federal Trade Commissioner appointed by Herbert Hoover. The Supreme Court tackled the case in Humphrey's Executor v. United States and ruled, essentially, that Congress does have the power to limit the president's power to fire officials of certain sorts of government agencies, such as the FTC and, presumably, also the Federal Reserve. I guess Trump could ask Congress to change the law to give him the power to fire the Fed chairman. I think that even if Trump asks for such a change, Congress is unlikely to deliver it, because it's actually pretty convenient for politicians to have a Fed chairman to blame for the economy or for stock market troubles. It helps the politicians avoid having to take personal responsibility for them. In that sense, the "independence" of the Fed serves a political function.