No do not remove the barrel. Go to Brownells and buy a broken case extractor for 223, or whatever other caliber you may need.
All you do is place in chamber close bolt then open it and the case will come out. Do not use anything else because they usually don't work, ie: over sized brush, cleaning jag packed with patches or any other home made remedy.
Cost of the proper tool used to be around $10.
If you are in a hurry your gun smith more than likely has one.
http://www.brownells.com/search/inde...ctor&ksubmit=y
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