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Your email should have a subject line that lets the professor know what to expect before they even open the email. Use a clear subject line, such as "Inquiry About Graduate Program" or "Questions From Potential Graduate Student." You can also include the specific research area you want to work in in the subject line. For instance, you could write "Inquiry into Graduate Work on Classical Music Theory." It is important to be formal when emailing a professor that you want to work with. Since you haven't met yet and you are hoping for a future relationship with this person, this is your chance to make a good first impression. Remember to use complete sentences and proper grammar. Also avoid using slang or informal language.  Begin your email with a formal salutation, such as "Dear Professor Smith." Conclude your email with a formal closing, such as "Sincerely" followed by your full name. The professor doesn't need to know all of the details of your life and every one of your qualifications for acceptance into grad school in this first email. However, you should tell them enough so that they know you are a viable candidate. This information usually includes a sentence or 2 about your background and undergraduate education, including the school you went to and the year you graduated.  You may also want to include a sentence about any special programs or academic projects you have completed. You want to strike a balance between letting the professor know what a motivated and good student you are and giving so many details that the professor loses interest. In general, hold back the minute details of your previous academic work on the first email and include them in later correspondence. When contacting a professor you need to tell them why you are interested in working with them and why their program is a good fit for you. If you have done your research on the graduate program in general and you have researched the professor's work, you can easily combine aspects of both of these things into a statement about why you are interested. This is a section that you can use to give the professor a few details about your interests. Tell them what you want to study and why you became interested in that subject. You can even include the name of a previous professor that you worked with studying this subject, especially if that professor will be writing you a letter of recommendation. In order to get a reply from the professor, you should ask specific questions that you would like answered. This ensures that the professor can easily draft a response letter to you and that your email is just the beginning of correspondence with the professor.  Do not ask questions that you can find the answers to online. This will be viewed as a waste of the professor's time. Some questions you can ask a potential professor include "Are you accepting new graduate students at this time?" and "Would you be willing to discuss the program with me either in person or over the phone?"
Write a clear, concise subject line. Use a professional structure. Begin by describing yourself and your qualifications for admission briefly. Show sincere interest in their program in the body of the email. Ask questions at the end of the email.