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Let $K$ be a field and let $f \in K[x]$ have degree $n \ge 1$. Then $f$ has a splitting field $L$ over $K$, and $[L : K] \le n!$.

Let $X$ be a real random variable with moment-generating function $M_X(t) = E[e^{tX}]$, and let $a \in \mathbb{R}$. Then $$P(X \ge a) \le \inf_{t > 0} e^{-ta} M_X(t).$$

Let $G$ be a group. Then $G$ is isomorphic to a subgroup of the symmetric group $\operatorname{Sym}(G)$. Explicitly, the map $g \mapsto \ell_g$, where $\ell_g(x) = gx$, is an injective group homomorphism from $G$ to $\operatorname{Sym}(G)$.

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $f : J \times U \to \mathbb{R}^n$ be continuous, and let $t_0 \in J$ and $x_0 \in U$. Let $\gamma : J \to U$ be a continuous curve. Then $\gamma$ is differentiable on $J$ with $\gamma'(t) = f(t, \gamma(t))$ for each $t \in J$ and $\gamma(t_0) = x_0$ if and only if $$\gamma(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds \quad \text{for each } t \in J.$$ At an endpoint of $J$ that belongs to $J$, the derivative is one-sided.

A glide reflection is a reflection in a hyperplane followed by a translation parallel to that hyperplane. In the plane, it is a reflection in a line followed by a shift along that line.

Let $K$ be a commutative ring, let $n \ge 1$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$. Let $S_n$ be the group of permutations of $\{1, \dots, n\}$, and let $\sgn(\sigma) \in \{1, -1\}$ be the sign of $\sigma$. Then the determinant of $A$ is $$\det A = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^{n} a_{i, \sigma(i)}.$$

The principal branch of a multivalued function is one branch of it that a standard convention chooses, see the definition. Its values are the principal values. An example is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$ with $\operatorname{Arg} z \in (-\pi, \pi]$.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded. Then $f$ is Riemann integrable on $[a, b]$ if and only if the following holds. For each $\varepsilon > 0$ there is a partition $P$ of $[a, b]$ with $U(f, P) - L(f, P) < \varepsilon$.

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces at each vertex. There are exactly five: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

Let $n$ be a natural number. Then the number of Dyck words of semilength $n$ is the Catalan number $C_n$.

A prime number is a natural number greater than $1$ whose only positive divisors are $1$ and itself. The definition of a prime number gives the precise form.

A vector-valued function is a function whose values are vectors, for example $f \colon D \to \mathbb{R}^m$ with $D \subseteq \mathbb{R}^n$ and $m \ge 2$. Such a function has $m$ component functions $f_1, \dots, f_m \colon D \to \mathbb{R}$, with $f(x) = (f_1(x), \dots, f_m(x))$ for each $x \in D$. An example is $r(t) = (\cos t, \sin t, t)$ for $t \in \mathbb{R}$, whose values trace a helix in the Euclidean space $\mathbb{R}^3$.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$. Then there is $c \in (a, b)$ with $f'(c) = 0$.

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be holomorphic on $0 < |z - c| < r$. Assume that $f$ has a simple pole at $c$, that is, a pole of order $1$. Then $$\operatorname{Res}(f, c) = \lim_{z \to c} (z - c) f(z),$$ where $\operatorname{Res}(f, c)$ is the residue of $f$ at $c$. In particular, let $f = g / h$, where $g$ and $h$ are holomorphic near $c$, $g(c) \neq 0$, $h(c) = 0$ and $h'(c) \neq 0$. Then $\operatorname{Res}(f, c) = \frac{g(c)}{h'(c)}$.

For each $z \in \mathbb{C}$, the complex exponential is complex differentiable at $z$, and $$\exp'(z) = \exp z.$$

A root of a polynomial $f$ over a ring $R$ is an element $c$ of $R$ with $f(c) = 0$. The definition of a root states this precisely.

Let $N$ be an integer and let $f \colon [N, \infty) \to \mathbb{R}$ be a decreasing function with $f(x) \ge 0$ for all $x \ge N$. Then the series $\sum_{n=N}^{\infty} f(n)$ converges if and only if the improper integral $\int_N^{\infty} f(x)\, dx$ converges.

Let $0 < a < 1$, and let $b$ be a positive odd integer with $a b > 1 + \frac{3}{2} \pi$. The Weierstrass function is $$W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x).$$ The function $W$ is continuous on $\mathbb{R}$, but it is not differentiable at any point. So a continuous function need not have a derivative anywhere.

Let $f$ be a real function on an open interval $(c - \delta, c + \delta)$ with $\delta > 0$, continuous at $c$ and differentiable at each $x \ne c$. If $f'(x) \le 0$ for $c - \delta < x < c$ and $f'(x) \ge 0$ for $c < x < c + \delta$, then $f$ has a local minimum at $c$. If the inequalities are reversed, then $f$ has a local maximum at $c$.

$$\cos \pi = -1, \qquad \sin \pi = 0.$$

Let $a$, $b$ and $c$ be real numbers with $(a, b) \neq (0, 0)$, and let $L = \{(x, y) \in \mathbb{R}^2 : a x + b y + c = 0\}$ be a line in the plane. Let $P = (x_0, y_0)$ be a point of $\mathbb{R}^2$. The distance from $P$ to $L$ is the smallest Euclidean distance $d(P, Q)$ for $Q \in L$. Then $$\operatorname{dist}(P, L) = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.$$

The preimage of a set under a function is the set of all inputs whose value is in that set.

Let $(X_n)_{n \ge 0}$ be a time-homogeneous Markov chain on a countable state space $S$. For $n \ge 0$, let $p^{(n)}_{ij} = P(X_{m+n} = j \mid X_m = i)$ be the $n$-step transition probabilities. Then for all $m, n \ge 0$ and $i, j \in S$, $$p^{(m+n)}_{ij} = \sum_{k \in S} p^{(m)}_{ik} \, p^{(n)}_{kj}.$$ In matrix form, $P^{(m+n)} = P^{(m)} P^{(n)}$, so $P^{(n)} = P^n$ for the transition matrix $P$.

Every nonempty set of natural numbers has a least element. That is, if $A \subseteq \mathbb{N}$ and $A \neq \emptyset$, then there is an $a$ in $A$ with $a \le x$ for each $x$ in $A$.

A well-order is a total order in which each nonempty subset has a least element.

There is no total order $\le$ on $\mathbb{C}$ that makes $\mathbb{C}$, with its usual addition and multiplication, an ordered field.

Let $G$ be a group and let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ if and only if $gN = Ng$ for each $g \in G$.

Public-key cryptography is a kind of cryptography in which each user has a pair of keys: a public key and a private key. Anyone can use the public key to encrypt a message or to check a signature. Only the owner of the private key can decrypt the message or make the signature. Its security depends on problems that are believed to be hard to solve, such as integer factorization or the discrete logarithm.

The law of a random variable $X$ is the probability distribution of its values: it gives each set $B$ of values the probability $P(X \in B)$. The node law of a random variable gives the precise definition.

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$. Then $$\det(cA) = c^n \det A,$$ where $cA$ is the scalar multiple and $\det$ is the determinant.

Let $G$ be a group and let $N$ be a normal subgroup of $G$. If $N$ and $G/N$ are solvable, then $G$ is solvable.

Let $X$ be a normal space and let $A, B \subseteq X$ be disjoint closed sets. Then there is a continuous map $f \colon X \to [0, 1]$ with $f(a) = 0$ for all $a \in A$ and $f(b) = 1$ for all $b \in B$.

Let $f \colon A \to B$ and $g \colon B \to C$ be injective functions. Then the composition $g \circ f \colon A \to C$ is injective.

A periodic continued fraction is an infinite simple continued fraction whose terms repeat a fixed block from some point on. The definition of a periodic continued fraction gives the precise form.

A generalised eigenvector of an endomorphism for $\lambda$ is a nonzero vector with a special property. Some power of the map minus $\lambda$ times the identity sends it to zero.

Let $\sim$ be an equivalence relation on a set $A$. Then the set $\{[a] \mid a \in A\}$ of equivalence classes is a partition of $A$.

A maximal solution of an ordinary differential equation is a solution that is not the restriction of a solution on a larger interval. Under a local Lipschitz condition, each initial value problem has exactly one maximal solution. See the definition.

Let $m$, $n$ and $r$ be natural numbers. Then $$\binom{m + n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r - k}.$$

A regular polygon is a polygon with all sides of the same length and all angles of the same size. Its vertices lie on one circle. Examples are the equilateral triangle and the square.

Let $(X, d)$ be a metric space. For all points $x, y, z \in X$, $$d(x, z) \le d(x, y) + d(y, z).$$