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The localization map $\iota :R\to S^{-1}R$ sends every element of $S$ to a unit. If a ring homomorphism $f:R\to T$ also sends every element of $S$ to a unit, there is a unique homomorphism $\tilde{f}:S^{-1}R\to T$ with $\tilde{f}(r/s)=f(r)f(s{)}^{-1}$ and $\tilde{f}\iota =f$.

A valuation ring is an integral domain $R$ such that, for every nonzero element $x$ of its fraction field $K$, either $x\in R$ or $x^{-1}\in R$. Equivalently, its ideals are totally ordered by inclusion.

Quantum teleportation transfers an unknown qubit using one shared Bell pair, a Bell-basis measurement by the sender, two classical bits, and a Pauli correction by the receiver.

Two parties can distinguish $|{\Phi }^{+}⟩$ from $|{\Phi }^{-}⟩$ by both measuring in the Hadamard basis and comparing outcomes: equal outcomes identify $|{\Phi }^{+}⟩$, while unequal outcomes identify $|{\Phi }^{-}⟩$.

The maximal abelian extension $K^{\mathrm{ab}}$ of a field $K$ is the compositum in a fixed separable closure of all finite abelian extensions of $K$.

The roots of unity in a finite extension $K/{\mathbb{Q}}_p$ form a finite group. Each element of their prime-to-$p$ part is a Teichmuller representative, while any $p$-power part lies among the principal units.

A p-adic unit is an invertible element of the ring ${\mathbb{Z}}_p$. For a p-adic integer $x$, this is equivalent to $v_p(x)=0$, $|x{|}_p=1$, or a nonzero initial p-adic digit $b_0$. A rational number $a/b$ is a p-adic unit when neither its reduced numerator nor its denominator is divisible by $p$.

Thus $3$ and $1/3$ are p-adic units at $p=2$, while $2$ is a p-adic integer with no inverse inside ${\mathbb{Z}}_2$. The modular inversion converges to a p-adic inverse construction computes the inverse of an integer unit to arbitrary finite precision.

For a finite extension $L/K$ of non-Archimedean local fields, the ramification index $e(L/K)$ is the index of the value group of $K$ in that of $L$. Normalized valuations satisfy $v_L{|}_K=e(L/K)v_K$.

The Newton polygon of a polynomial over a valued field is the lower convex hull of the points formed from coefficient indices and valuations. Its slopes determine the valuations of roots, counted with multiplicity.

For a discrete valuation ring with uniformizer $\pi$, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by $\pi$ and whose constant coefficient is not divisible by ${\pi }^2$. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

A finite extension of local fields is unramified when its ramification index is one and its residue-field degree equals the field degree. A local field has a unique unramified extension of each positive degree inside a fixed algebraic closure.

The Teichmuller representative of a nonzero residue class $\bar{a}\in {\mathbb{F}}_q^{\ast }$ is the unique lift $\omega$ satisfying ${\omega }^{q-1}=1$ and $\omega \equiv a$ modulo the maximal ideal.

The $p$-adic logarithm is the convergent series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ on a sufficiently small group of principal units.
It is locally inverse to the p-adic exponential function.

The Laurent series field $k((t))$ consists of formal series ${\sum }_{n\ge N}{a}_n{t}^n$ with finitely many negative-degree terms. Its $t$-adic valuation is the least exponent with nonzero coefficient.

A Lipschitz bound is any valid inequality of the displayed form with a finite constant $L$. It need not use the least Lipschitz constant. For $f(x)=3x-2$, both $L=3$ and $L=5$ are valid, but $L=2$ fails. The tighter bound gives a sharper error estimate: input error at most $0.01$ guarantees output error at most $0.03$ using $L=3$, compared with the valid but weaker $0.05$ using $L=5$. The bound constrains errors; it does not state that every pair achieves that amplification.

An entire function bounded by a polynomial in the radius is itself a polynomial, by Cauchy estimates.

If an entire function satisfies $|h(z)|\le |\mathrm{Re}z{|}^{-1/2}$ away from the imaginary axis, then $h=0$. Cauchy's formula on $|z|=R$ bounds every Taylor coefficient by a constant timeswhich tends to zero as $R\to \infty$.

The image of every nonconstant entire function is dense in $\mathbb{C}$. If a disc about $w$ were omitted, then $1/(f-w)$ would be bounded and entire, so Liouville's theorem would make $f$ constant.

A Riccati equation is a first-order equation quadratic in the unknown function.

Every convergent series of positive rational terms has a subseries whose rational partial sums $p_j/q_j$ satisfyChoose each new term small enough to enforce all previous tail bounds. Separation of reduced fractions first makes $\alpha$ irrational, and the Liouville approximation theorem then rules out every finite algebraic degree.