degree of product of two polynomials equals to sum of degree   ### poly_degree natural_description
$\deg f*g = \deg f + \deg g$  ### poly_degree latex_formula
`(f * g).degree = f.degree + g.degree`  ### poly_degree lean_term
Every finite domain is a field. ### finite_domain natural_description
Wedderburn's little theorem ### finite_domain theorem_name
the universal property of the localization of ring ### ring_local natural_description
Given ring homomorphism $f : A \to B$, if the image of $S \subset A$ is invertible in $B$, then $f$ factors through $S^{-1}A$ ### ring_local latex_formula
derivative of sum = sum of derivatives ### derivative_sum natural_description
$f'(x) + g'(x) = (f + g)'(x)$ ### derivative_sum latex_formula
`deriv f x + deriv g x = deriv (f + g) x` ### derivative_sum lean_term
Integration by substitution on an interval ### integral_substitution natural_description
$\int_a^b g(f(x)) \cdot f(x) dx = \int_{f(a)}^{f(b)} g(y) dy$ ### integral_substitution latex_formula
Pi is greater than 3 ### pi_3 natural_description
$\pi > 3$ ### pi_3 latex_formula
`Real.pi > 3` ### pi_3 lean_term
Order of an element divides the order of the group ### group_order natural_description
$\text{Ord}(x) \mid |G|, \forall x \in G$ ### group_order latex_formula
If $G$ is a group, $g \in G$, `orderOf g ∣ G.card` ### group_order lean_term
The Schur-Zassenhaus Theorem ### Schur_Zassenhaus theorem_name
Schur-Zassenhaus ### Schur_Zassenhaus theorem_name
Let $G$ be a finite subgroup, $H$ be a normal subgroup of $G$, then there exist a subgroup $K$ such that $K$ is the complement of $H$ in $G$. ### Schur_Zassenhaus natural_description
the number of Sylow $p$ subgroup equals $1 \mod p$ ### sylow_third latex_formula
Sylow’s third theorem ### sylow_third theorem_name
`Fintype.card (Sylow G p)` equals $1 \mod p$ ### sylow_third lean_term
The trace of the product of two matrices is independent of order of multiplication ### trace_swap natural_description
tr($AB$) = tr($BA$) ### trace_swap natural_description
`(A * B).trace = (B * A).trace` ### trace_swap lean_term
All eigenvalues of a self-adjoint matrix are real ### eigenvalue_sym natural_description
If $A$ is symmetric, eigenvalues of $A$ are real. ### eigenvalue_sym natural_description
expansion of determinant of 2*2 matrix ### 2d_determinant natural_description
$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a*c-b*d $ ### 2d_determinant latex_formula
If $A$ is a $2*2$ matrix, $\det A = a_{11}*a_{22} - a_{12}*a_{21}$. ### 2d_determinant latex_formula
For integers $a,b,c$, if $a$ divides $b$ and $c$, then $a$ divides the greatest common divisor of $b$ and $c$ ### divide_gcd natural_description
For $a, b, c \in \mathbb{Z}$, $a \mid b$ and $a \mid c$ implies $a \mid \text{gcd}(b, c)$ ### divide_gcd latex_formula
`(a b c : ℤ) : a ∣ b → a ∣ c → a ∣ gcd b c` ### divide_gcd lean_term
fundamental identity of ramification and inertia ### ramify_inertia theorem_name
$\Sum e_i f_i =$ extension degree, where $e_i$ is ramification index and $f_i$ is inertia degree ### ramify_inertia latex_formula
For any Galois extension, the fixed field of fixing subgroup of any intermediate field is itself. ### Galois_extension natural_description
Let $L/K$ be a Galois extension, $F$ be an intermidiate field, then $L^{\{\sigma \in \text{Gal}(L/K) | \sigma x = x , \forall x \in F \} } = F$ ### Galois_extension latex_formula
any set is not an element of itself in ZFC ### ZFC_set natural_description
$x \notin x$ ### ZFC_set latex_formula
`¬ x ∈ x` ### ZFC_set lean_term
if there exist injective maps of sets from $A$ to $B$ and from $B$ to $A$, then there exist bijective map between $A$ and $B$ ### bijective_exist natural_description
If there exist $f : A \to B$ injective, $g : A \to B$ injective, then there exist $h : A \to B$ bijective. ### bijective_exist latex_formula
Schroeder Bernstein theorem ### bijective_exist theorem_name
`{f : A → B} {g : B → A} (hf : f.Injective) (hg : g.Injective) : ∃ h, h.Bijective` ### bijective_exist lean_term
if $p$ implies $q$, then not $q$ implies not $p$ ### modus_tollens natural_description
modus tollens ### modus_tollens theorem_name
$(p \to q) \to (\not q \to \not p)$ ### modus_tollens latex_formula
`(p → q) → (¬ q → ¬ p)` ### modus_tollens lean_term