The Doctor of Philosophy (Ph.D.) is a research degree designed to prepare students to discover, integrate, and apply knowledge as well as to communicate and disseminate it. Students are required to enroll in 9 credit-hours per semester. While the Ph.D. degree formally requires a minimum of 30 semester hours of advanced coursework, including a minimum of six dissertation hours, high-performing students normally require five to six years of full-time enrollment (9-12 semesters) to complete requirements of the Ph.D. degree. It is quite exceptional (applies to less than 1% of students, and only in unusual circumstances) that a student is able to complete the requirements in less than 9 semesters. In particular, completion of a research program leading to a dissertation worthy of the Ph.D. degree usually requires at least three years of work, taking the form both of conference courses with an agreed academic advisor (before formal passage to candidacy) and as dissertation hours (post-candidacy). While the overall degree generally requires five to six years, the distribution of the coursework and dissertation components of the degree varies considerably. Among other factors, it depends on the mathematical preparation of the student on entry.