Fairleigh Dickinson University (Part 1 - Professor Dolbin)
(8:00 - 12:00)
Today I further studied the intricate structure of Galois groups. I learned the Fundamental Theorem of Galois Theory, which says that there is a direct correspondence between extensions of fields and subgroups of Galois groups. I also became used to drawing Galois correspondence diagrams, which shows this relationship pictorially. It’s amazing how the two types of mathematical structures, which seem entirely different at first, can be related in such a simple way. Moreover, I learned that all of this can be further related to the one type of groups called the symmetric groups, which are just the set of ways to permute a set of objects. I see this in two ways—mathematicians have either shown that these seemingly complicated structures in mathematics boil down to the simple group of rearrangements, or that the seemingly simple symmetric group is, in fact, monstrous and obscenely complex. I guess it depends on whether one is a pessimist or an optimist. Nonetheless, after studying these structures, I moved on to the proof for why there is no generic solution to the cubic equation, which utilizes the material that I have learned so far. In addition, under the guidance of Professor Dolbin, I have learned several interesting side details along the way. Nonetheless, by the end of the day I was able to finish yet another chapter of the text, thus taking a gigantic step towards my algebraic research project.
(Part 2 - Professor Farag)
(1:00 - 3:00)
Next, in Professor Farag’s class, we further studied rings. We covered a special case of rings called ideals, which have a unique "absorption" property. Afterwards, we discussed some more practice problems and went over the last problem set, for which I had already written up solutions. Today I gained further knowledge about abstract algebra, and this class has been helpful especially for the first part of my internship, as one supports and adds more information to the other.
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