Proving J[i] is a Euclidean ring.

Today we’ll try to realign our intuition with the standard textbook proof of “ is a Euclidean ring”. denotes the set of all complex numbers of the form where and are integers. . Let us take two complex numbers . Let and , where and . Then if there exists a complex number such thatContinue reading “Proving J[i] is a Euclidean ring.”

Euclidean rings and prime factorization

Now we will talk about the factorization of elements in Euclidean rings. On pg.146 of “Topics in Algebra” by Herstein, it says: “Let be a Euclidean ring. Then every element in is either a unit in or can be written as the product of a finite number of prime elements in .” This seems elementary.Continue reading “Euclidean rings and prime factorization”

Euclidean rings and generators of ideals

This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein. In a Euclidean ring, for any two elements such that . Also, there exists a function such that . We also know that the element with the lowest d-value generates the whole ring . The proof ofContinue reading “Euclidean rings and generators of ideals”

Integral domains and characteristics

Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points. An integral domain is a commutative ring with the property that if and , then . Hence, if , then or (or both). The characteristic of an integral domain is the lowest positive integer such that .Continue reading “Integral domains and characteristics”

Ordinals- just what exactly are they?!

If ordinals have not confused you, you haven’t really made a serious attempt to understand them. Let me illustrate this. If I have 5 fruits (all different) and 5 plates (all different), then I can bijectively map the fruits to plates. However, I arrange the fruits or plates, I can still bijectively map them. Let’sContinue reading “Ordinals- just what exactly are they?!”

Why substitution works in indefinite integration

Let’s integrate . We know the trick: substitute for . We get . Substituting into the original equation, we get . Let us assume remains positive throughout the interval under consideration. Then we get the integral as or . I have performed similar operations for close to five years of my life now. But IContinue reading “Why substitution works in indefinite integration”

Fermat’s Last Theorem

When in high school, spurred by Mr. Scheelbeek’s end-of-term inspirational lecture on Fermat’s Last Theorem, I tried proving the same for…about one and a half long years! For documentation purposes, I’m attaching my proof. Feel free to outline the flaws in the comments section. Let us assume FLT is true. i.e. . We know (Continue reading “Fermat’s Last Theorem”

Binomial probability distribution

What exactly is binomial distribution? Q. A manufacturing process is estimated to produce nonconforming items. If a random sample of the five items is chosen, find the probability of getting two nonconforming items. Now one could say let there be items. Then the required probability woud be . In what order the items are chosenContinue reading “Binomial probability distribution”

Continuous linear operators are bounded.: decoding the proof, and how the mathematician chances upon it

Here we try to prove that a linear operator, if continuous, is bounded. Continuity implies: for any for We want the following result: , where is a constant, and is any vector in . What constants can be construed from and , knowing that they are prone to change? As is a linear operator, isContinue reading “Continuous linear operators are bounded.: decoding the proof, and how the mathematician chances upon it”

Linear operators mapping finite dimensional vector spaces are bounded,

Theorem: Every linear operator , where is finite dimensional, is bounded. Proof where . What we learn from here is where . Similarly, where Another proof of the assertion is which is a constant. Note: why does this not work in infinite dimensional spaces? Because the max and min of and might not exist.