Yellow phlegm is usually a sign of bronchitis and is common following a cold or sinus problem.
Most cases will clear up with rest and by drinking fluids. You may want to consider taking an over the counter expectorant if the phlegm becomes thick. Try to maintain humidity in the home also.
You should see a doctor if you can't rest because the cough becomes bothersome, if you have a persistent fever, or develop wheezing.
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Many doctors do provide antibiotics if a bacterial infection is suspected - when the symptoms recur, are prolonged or if fever develops. This is also common if you smoke or have underlying illnesses and it's done to prevent an infection.
If this is the case, that is if you smoke or have underlying lung disease or poor immune function, then your doctor may really want you on an antibiotic, but again, bronchitis is usually caused by a virus.
Bronchitis is usually the result of a viral infection, and antibiotics do not effectively treat viral infections. Feel free to contact me if you have any other questions at all.
I hope this clears things up.