## A few mathematical study skills... Reading Theorems

by Ashley Reiter Ahlin

In almost any advanced math text, theorems, their proofs, and motivation for them make up a significant portion of the text. The question then arises, how does one read and understand a theorem properly? What is important to know and remember about a theorem?

A few questions to consider are:

• What kind of theorem is this? Some possibilities are:
• A classification of some type of object (e.g., the classification of finitely generated abelian groups)
• An equivalence of definitions (e.g., a subgroup is normal if, equivalently, it is the kernel of a group homomorphism, or its left and right cosets coincide)
• An implication between definitions (e.g., any PID is a UFD)
• A proof of when a technique is justified (e.g., the Euclidean algorithm may be used when we are in a Euclidean domain)
• Can you think of others?
• What's the content of this theorem? I.e., are there some cases in which it is trivial, or in which we've already proven it?
• Why are each of the hypotheses needed? Can you find a counterexample to the theorem in the absence of each of the hypotheses? Are any of the hypotheses unnecessary? Is there a simpler proof if we add extra hypotheses?
• How does this theorem relate to other theorems? Does it strengthen a theorem we've already proven? Is it an important step in the proof of some other theorem? Is it surprising?
• What's the motivation for this theorem? What question does it answer?