## Ah. Primes, irreducible elements, and Number Theory

Proving that primes are irreducible in any integral domain is simple. Assume that a prime is not irreducible. Then , where and are not units. Now we know that . Therefore, . Let . Then , where . Now we have . This implies , which is a contradiction as cannot be a unit. ButContinue reading “Ah. Primes, irreducible elements, and Number Theory”

Something that has confused me for long is the condition for injectivity for a homomorphism. The condition is that the kernel should just be the identity element. I used to think that maybe this condition for injectivity applies to all mappings, and wondered why I hadn’t come across this earlier. No. This condition applies onlyContinue reading

## Note to self.

may not be equal to . Topology.

## Gauss’s lemma (polynomials)

I have long interpreted Gauss’s lemma to mean that if a polynomial with integral coefficients has a rational root, that root has to be an integer. This is incorrect. For example, take the polynomial . It has integral coefficients. However, it does not have an integral root. It has a rational root; namely . Gauss’sContinue reading “Gauss’s lemma (polynomials)”

## Small note

One might have wondered why contains only linear bounded operators, and not linear operators of any and every kind. This has a very specific reason. Unless we fix , we cannot construct a cauchy sequence of linear operators. And we do not want to fix , as we want to define the limit for everyContinue reading “Small note”

## The strange difference between “divergent sequences” in real analysis and abstract algebra

I have been working on Commutative Algebra. A lot of the initial proofs that I’ve come across use Zorn’s lemma. The statement of Zorn’s lemma is simple enough (which I have blogged about before): Suppose a partially ordered set has the property that every chain (i.e. totally ordered subset) has an upper bound in .Continue reading “The strange difference between “divergent sequences” in real analysis and abstract algebra”