Algebraic Elements and Transcendental Elements
Proposition 1. Let $E=F(u)$, where $u$ is transcendental over $F$. If $K\neq F$ and $K$ is an intermediate field of $E/F$, then $u$ is algebraic over $K$.
Theorem 1. Let $E/F$ be a field extension, and let $a$ be algebraic over $F$. Then
$$
[F(a):F]=\deg f(x),
$$
where $f(x)$ is the minimal polynomial of $a$ over $F$.
Automorphisms of the Rational Field and the Real Field
Proposition 1. The only automorphism of the rational field $\mathbb{Q}$ is the identity automorphism.
Definition. Let $R$ be a commutative ring with identity. If $I$ and $J$ are two ideals of $R$ such that
$$
I+J=R,
$$
then $I$ and $J$ are said to be coprime.
Proposition 1. Let $I$ and $J$ be ideals of a ring $R$. Then:
$IJ \subseteq I$ and $IJ \subseteq J$.
$IJ \subseteq I \cap J \subseteq I+J$.
Rings Without Nontrivial One-Sided Ideals
First, let us review the notion of an ideal generated by an element.
On the Invertibility of $1-ab$ in a Unital Ring
Theorem. Let $R$ be a ring with identity, and let $a,b \in R$. Then
$$
1-ab \text{ is invertible } \iff 1-ba \text{ is invertible}.
$$
On the Problem of Elements in a Ring Having More than One One-Sided Inverse
Lemma 1. Let $R$ be a ring with identity, and let $a \in R$. Suppose that $a$ has a left inverse in $R$. Then the following are equivalent:
$a$ has more than one left inverse.
$a$ is not a unit.
$a$ is a right zero divisor of some element.
On the Orders of Elements in Direct Product Groups
Proposition 1. Let $a$ be an element of order $m$ in a group $G$, and let $b$ be an element of order $n$ in a group $G$. Then the element $(a,b)$ in the direct product $G \times G$ has order $[m,n]$.
Some Simple Applications of the Sylow Theorems
Proposition 1. Any group $G$ of order $20449 = 11^2 \cdot 13^2$ must be abelian.