A generalization of “all members of a simple field extension of F are algebraic over F”

The following is a powerful powerful theorem: Let be a simple field extension of field , where is algebraic over . Then every is also algebraic over . Also, . The proof I think is most brilliant. I am not going to provide you with a proof here. I am just going to try andContinue reading “A generalization of “all members of a simple field extension of F are algebraic over F””

Trigonometric substitution in integration

Why trigonometric substitution in integration: this is something that puzzled me and made me hate differentiation/integration during my IITJEE preparation. A lot of the techniques that we learned were based on algorithmic memorization rather than a feel for what really was happening. Thankfully, I have come to a college that requires 0% attendance, so thatContinue reading “Trigonometric substitution in integration”

Racism and path homotopy

I recently had to face some racism on math.stackexchange.com for asking some seemingly obvious question on path homotopy. In retrospect, I could have thought through the concept myself. However, I don’t think I would have done so as quickly had I not been angry about the racism. This is what I was trying to prove:Continue reading “Racism and path homotopy”

Why fundamental groups are defined only for loops- better explained than in Munkres’ Topology.

I want to point out why fundamental groups are defined for loops, and not path homotopy classes. This is something that Munkres’ Topology does not do a good job of explaining. Munkres says that we cannot define groups on path homotopy classes because for some pair and , may not be defined. This is becauseContinue reading “Why fundamental groups are defined only for loops- better explained than in Munkres’ Topology.”

Reaching infinity- Lobbying for Axiom A

This is a post on one aspect of the infinitely muddled and confusing (confused?) concept of cardinality. Take the product and box topologies of . The product topology is first countable, whilst the box topology is not. A good discussion is given in Munkres. I will discuss this with an analogy. Let us take aContinue reading “Reaching infinity- Lobbying for Axiom A”

Zorn’s lemma

This is another rant on Zorn’s lemma. And hopefully the final one. It is something that has puzzled me for more than a year now! In a partially ordered set , let every chain have an upper bound. Why is it not obvious that has a maximal element? The whole concept depends on the followingContinue reading “Zorn’s lemma”

Why isomorphisms

I often wondered why isomorphisms were important. And if you haven’t done the same, maybe you have studied Abstract Algebra rather passively. We shall explore this question today. We know (or assume for the moment) that is isomorphic to . The mappings are . You can verify all this for yourself. Now let us supposeContinue reading “Why isomorphisms”

The minimal polynomial of a transformation

For a linear transformation , the minimal polynomial is a very interesting polynomial indeed. I will discuss its most interesting property below: Let be the minimal polynomial of transformation . Then for and . You may be shocked (I hope Mathematics has that kind of effect on you :P). Why this is possible is thatContinue reading “The minimal polynomial of a transformation”