有理数的可数性

一种求解一阶和双曲二阶偏微分方程 (PDE) 的技术。该方法将偏微分方程 (PDE) 简化为一系列沿着特定曲线（称为“特征”）的常微分方程 (ODE)。沿着这些曲线，PDE 会简化，从而可以通过对 ODE 组进行积分来求得解。该方法对于涉及传输和波传播的问题尤其有效。

A direct corollary of the fundamental theorem of algebra is that any non-constant polynomial with real coefficients can be factored into a product of linear factors and irreducible quadratic factors, all with real coefficients. The linear factors correspond to the real roots, while the irreducible quadratic factors correspond to pairs of complex conjugate roots [latex]a \pm bi[/latex].

A transcendental number is a real or complex number that is not algebraic, meaning it is not a root of any non-zero polynomial equation with integer (or rational) coefficients. All transcendental numbers are irrational, but not all irrational numbers are transcendental (e.g., [latex]\sqrt{2}[/latex] is irrational but algebraic, as it is a root of [latex]x^2 - 2 = 0[/latex]).

Hilbert's Nullstellensatz (German for "theorem of zeros") establishes a fundamental correspondence between geometry and algebra. It states that for an algebraically closed field [latex]k[/latex], if a polynomial [latex]p[/latex] vanishes on the zero-set of an ideal [latex]I[/latex], then some power of [latex]p[/latex] must belong to [latex]I[/latex]. Formally, [latex]I(V(I)) = \sqrt{I}[/latex], the radical of [latex]I[/latex].

An affine variety is the set of points in an affine space whose coordinates are the common zeros of a finite set of polynomials. For a set of polynomials [latex]S = \{f_1, \dots, f_k\}[/latex] in a polynomial ring [latex]k[x_1, \dots, x_n][/latex], the corresponding affine variety is [latex]V(S) = \{x \in k^n | f(x) = 0 \text{ for all } f \in S\}[/latex]. It is a central object of study in classical algebraic geometry.

Bézout's theorem is a fundamental statement in intersection theory. It asserts that the number of intersection points of two plane algebraic curves of degrees [latex]m[/latex] and [latex]n[/latex] is exactly [latex]mn[/latex], provided that one works in a projective plane over an algebraically closed field, counts points with multiplicity, and includes points at infinity where parallel asymptotes meet.

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This guarantees that the field of complex numbers is algebraically closed, meaning polynomial equations that cannot be solved in real numbers can be solved in complex numbers. For a polynomial [latex]p(z) = a_n z^n + \dots + a_1 z + a_0[/latex], there exists a [latex]z_0 in \mathbb{C}[/latex] such that [latex]p(z_0) = 0[/latex].

This theorem states that every integer greater than 1 is either a prime number or can be uniquely represented as a product of prime numbers, disregarding the order of the factors. For example, [latex]1200 = 2^4 \times 3^1 \times 5^2[/latex]. This unique factorization is a cornerstone of number theory, providing a fundamental multiplicative structure for the integers.

The canonical representation, or standard form, of a positive integer [latex]n[/latex] is its unique prime factorization written as a product of prime powers with the primes in increasing order. For any integer [latex]n > 1[/latex], it can be written as [latex]n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}[/latex], where [latex]p_1 < p_2 < \cdots < p_k[/latex] are prime numbers and the exponents [latex]a_i[/latex] are positive integers.

与柯西初值问题相关的偏微分方程的基本存在唯一性定理。该定理指出，如果偏微分方程和初始条件是“解析的”（可以用收敛幂级数表示），则在初始曲面的邻域中存在唯一解析解。它提供了局部存在性的保证，但并未解决全局行为或适定性问题。

For a prime number [latex]p[/latex], the p-adic numbers form an extension of the rational numbers that is topologically different from the real numbers. While real numbers are a completion of [latex]\mathbb{Q}[/latex] with respect to the usual absolute value metric, the p-adic numbers are the completion of [latex]\mathbb{Q}[/latex] with respect to the p-adic metric, where numbers are "small" if they are divisible by a high power of [latex]p[/latex].

Sheaf cohomology is a central tool in modern algebraic geometry for studying global properties of geometric spaces. For a sheaf [latex]\mathcal{F}[/latex] on a space [latex]X[/latex], the cohomology groups [latex]H^i(X, \mathcal{F})[/latex] are vector spaces whose dimensions provide important invariants. The group [latex]H^0[/latex] represents global sections, while higher groups [latex]H^i[/latex] for [latex]i > 0[/latex] measure the obstructions to patching together local sections into a global one.