Abstract Algebra IBL Wiki

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I'm going to be creating material for a course on the second half of Abstract Algebra. It will assume that students know the basic definitions and ideas about groups, especially automorphisms. Obviously I don't have a class, so no one will be using this any time soon. But if anyone wants to take problems from me that would be great.


I'm going to start with a review that preps for Galois Theory. Then I will introduce Galois theory but might not even use that word. Then I will introduce Category Theory. I'm not going to be concerned if they learn the definitions that are critical to Galois and Category theory, but it should give good practice using rings and fields, as well be engaging. 


=====Review=====
[[Problem:$S_X$ is a Group]]
{{page>Problem:$S_X$ is a Group#problem}}

[[Problem:$a+b\sqrt{2}$]]
{{page>Problem:$a+b\sqrt{2}$#problem}}

[[Problem:Show that $Q\[\alpha\]$ is a Field]]
{{pabe>Problem:Show that $Q\[\alpha\]$ is a Field}}






[[Problem:Binomial Theorem]]
{{pabe>Problem:Binomial Theorem}}

=====Introduction to Galois Theory=====


=====Introduction to Category Theory=====
We've seen in Group Theory how we can add, subtract, multiply, and sometimes divide things that aren't numbers. In other words Group Theory has let to us being able to add apples and oranges. Category Theory goes one step further, and lets us get an apple from an orange tree. Or in other words, Category Theory lets us get groups from rings, or fields, or topological spaces. Or it lets us get topological spaces from groups or rings and so forth. Category Theory uses functions to map from one object to another. And the objects can be from different categories (ie. one could be a ring, and the other a group.) Several common categories are the category or rings, the category of groups, the category of fields, the category of sets and the category of topological spaces.

[[Problem:Endomorphisms]]
{{pabe>Problem:Endomorphisms}}


[[Problem:Group Ring of G over R]]
{{pabe>Problem:Group Ring of G over R}}


[[Problem:Real Quaternions]]
{{pabe>Problem:Real Quaternions}}



[[Problem:Endomorphisms]]
{{pabe>Problem:Endomorphisms}}

[[Problem:Endomorphisms]]
{{pabe>Problem:Endomorphisms}}

[[Problem:Endomorphisms]]
{{pabe>Problem:Endomorphisms}}

[[Problem:Endomorphisms]]
{{pabe>Problem:Endomorphisms}}

=====Review=====

{{tag>tarafife}}
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