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

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)”

A generalisation of Gram-Schmidt’s orthonogonalisation process

I just read up about the Gram-Schmidt orthogonalization process. Say we have as an orthonormal basis for a subspace. Now let for ANY , where is the vector space. The set is orthonormal. $a$ needn’t be a vector from , where is spanned by . We require to be from only because we want toContinue reading “A generalisation of Gram-Schmidt’s orthonogonalisation process”

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”