A monotonic function whose points of discontinuity form a dense set

April 30, 2017 Jean-Pierre Merx Leave a comment

Consider a compact interval $[a,b] \subset \mathbb R$ with $a \lt b$ . Let’s build an increasing function $f : [a,b] \to \mathbb R$ whose points of discontinuity is an arbitrary dense subset $D = \{d_n \ ; \ n \in \mathbb N\}$ of $[a,b]$ , for example $D = \mathbb Q \cap [a,b]$ .

Let $\sum p_n$ be a convergent series of positive numbers whose sum is equal to $p$ and define $\displaystyle f(x) = \sum_{d_n \le x} p_n$ .

$f$ is strictly increasing

For $a \le x \lt y \le b$ we have $f(y) – f(x) = \sum_{x \lt d_n \le y} p_n \gt 0$ as the $p_n$ are positive and dense so it exists $p_m \in (x, y]$ .

$f$ is right-continuous on $[a,b]$

We pick-up $x \in [a,b]$ . For any $\epsilon \gt 0$ is exists $N \in \mathbb N$ such that $0 \lt \sum_{n \gt N} p_n \lt \epsilon$ . Let $\delta > 0$ be so small that the interval $(x,x+\delta)$ doesn’t contain any point in the finite set $\{p_1, \dots, p_N\}$ . Then $0 \lt f(y) – f(x) \le \sum_{n \gt N} p_n \lt \epsilon,$ for any $y \in (x,x+\delta)$ proving the right-continuity of $f$ at $x$ . Continue reading A monotonic function whose points of discontinuity form a dense set →

Algebra

A normal extension of a normal extension may not be normal

April 23, 2017 Jean-Pierre Merx Leave a comment

An algebraic field extension $K \subset L$ is said to be normal if every irreducible polynomial, either has no root in $L$ or splits into linear factors in $L$ .

One can prove that if $L$ is a normal extension of $K$ and if $E$ is an intermediate extension (i.e., $K \subset E \subset L$ ), then $L$ is a normal extension of $E$ .

However a normal extension of a normal extension may not be normal and the extensions $\mathbb Q \subset \mathbb Q(\sqrt{2}) \subset \mathbb Q(\sqrt[4]{2})$ provide a counterexample. Let’s prove it.

As a short lemma, we prove that a quadratic extension $k \subset K$ , i.e. an extension of degree two is normal. Suppose that $P$ is an irreducible polynomial of $k[x]$ with a root $a \in K$ . If $a \in k$ then the degree of $P$ is equal to $1$ and we’re done. Otherwise $(1, a)$ is a basis of $K$ over $k$ and there exist $\lambda, \mu \in k$ such that $a^2 = \lambda a +\mu$ . As $a \notin k$ , $Q(x)= x^2 – \lambda x -\mu$ is the minimal polynomial of $a$ over $k$ . As $P$ is supposed to be irreducible, we get $Q = P$ . And we can conclude as $Q(x) = (x-a)(x- \lambda +a).$

The entensions $\mathbb Q \subset \mathbb Q(\sqrt{2})$ and $\mathbb Q(\sqrt{2}) \subset \mathbb Q(\sqrt[4]{2})$ are quadratic, hence normal according to previous lemma and $\sqrt[4]{2}$ is a root of the polynomial $P(x)= x^4-2$ of $\mathbb Q[x]$ . According to Eisenstein’s criterion $P$ is irreducible over $\mathbb Q$ . However $\mathbb Q(\sqrt[4]{2}) \subset \mathbb R$ while the roots of $P$ are $\pm \sqrt[4]{2}, \pm i \sqrt[4]{2}$ and therefore not all real. We can conclude that $\mathbb Q \subset \mathbb Q(\sqrt[4]{2})$ is not normal.

Algebra

The image of an ideal may not be an ideal

April 18, 2017 Jean-Pierre Merx Leave a comment

If $\phi : A \to B$ is a ring homomorphism then the image of a subring $S \subset A$ is a subring $\phi(A) \subset B$ . Is the image of an ideal under a ring homomorphism also an ideal? The answer is negative. Let’s provide a simple counterexample.

Let’s take $A=\mathbb Z$ the ring of the integers and for $B$ the ring of the polynomials with integer coefficients $\mathbb Z[x]$ . The inclusion $\phi : \mathbb Z \to \mathbb Z[x]$ is a ring homorphism. The subset $2 \mathbb Z \subset \mathbb Z$ of even integers is an ideal. However $2 \mathbb Z$ is not an ideal of $\mathbb Z[x]$ as for example $2x \notin 2\mathbb Z$ .

Analysis

A function whose Maclaurin series converges only at zero

April 11, 2017 Jean-Pierre Merx Leave a comment

Let’s describe a real function $f$ whose Maclaurin series converges only at zero. For $n \ge 0$ we denote $f_n(x)= e^{-n} \cos n^2x$ and $f(x) = \sum_{n=0}^\infty f_n(x)=\sum_{n=0}^\infty e^{-n} \cos n^2 x.$ For $k \ge 0$ , the $k$ th-derivative of $f_n$ is $f_n^{(k)}(x) = e^{-n} n^{2k} \cos \left(n^2 x + \frac{k \pi}{2}\right)$ and $\left\vert f_n^{(k)}(x) \right\vert \le e^{-n} n^{2k}$ for all $x \in \mathbb R$ . Therefore $\displaystyle \sum_{n=0}^\infty f_n^{(k)}(x)$ is normally convergent and $f$ is an indefinitely differentiable function with $f^{(k)}(x) = \sum_{n=0}^\infty e^{-n} n^{2k} \cos \left(n^2 x + \frac{k \pi}{2}\right).$ Its Maclaurin series has only terms of even degree and the absolute value of the term of degree $2k$ is $\left(\sum_{n=0}^\infty e^{-n} n^{4k}\right)\frac{x^{2k}}{(2k)!} > e^{-2k} (2k)^{4k}\frac{x^{2k}}{(2k)!} > \left(\frac{2kx}{e}\right)^{2k}.$ The right hand side of this inequality is greater than $1$ for $k \ge \frac{e}{2x}$ . This means that for any nonzero $x$ the Maclaurin series for $f$ diverges.

Math Counterexamples

Monthly Archives: April 2017

A monotonic function whose points of discontinuity form a dense set

$f$ is strictly increasing

$f$ is right-continuous on $[a,b]$

A normal extension of a normal extension may not be normal

The image of an ideal may not be an ideal

A function whose Maclaurin series converges only at zero

Mathematical exceptions to the rules or intuition

ff is strictly increasing

ff is right-continuous on [a,b][a,b]

The Hamilton’s quaternions group is not a semi-direct product

Mathematical exceptions to the rules or intuition

$f$ is strictly increasing

$f$ is right-continuous on $[a,b]$