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”

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”