Intereting Posts

Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$
compound of gamma and exponential distribution
Number of $\sigma$ -Algebra on the finite set
Guessing one root of a cubic equation for Hit and Trial
Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.
Intersection of a properly nested sequence of convex sets , if nonempty and bounded , can never be open?
Why can't a set have two elements of the same value?
Do continuous linear functions between Banach spaces extend?
Sums of Consecutive Cubes (Trouble Interpreting Question)
How to determine in polynomial time if a number is a product of two consecutive primes?
Infinite powering by $i$
Easy partial fraction decomposition with complex numbers
Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.
What is the trellis diagram for a linear block code?
If $f(x)$ is uniformly continuous for $x \ge 0$, then it is bounded above by a straight line.

For example, how can I show that $\mathbb{Q}$ is the fraction field of $\mathbb{Z}$? Or that $\mathbb{C}$ is the fraction field of $\mathbb{R}$?

I understand that $\mathbb{Z}$ is a subring of $\mathbb{Q}$ & each r in $\mathbb{Q}$ can be written as a fraction r = a/b with a,b in $\mathbb{Z}$ and no proper subfield of $\mathbb{Q}$ has that property. But is there some general way to show this for the other number systems?

- Describe the units in $\mathbb{Z}$
- Intermediate fields of a finite field extension that is not separable
- What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?
- Ring homomorphism from $\mathbb Z$ to $\mathbb Z$ is always identity or $0$
- Multivariable Gauss's Lemma
- Computing $\operatorname{Tor}_1^R(R/I,R/J)$

- Base-b Representation of an Integer: Why can I make the assumption about number of terms in expansions in uniqueness part?
- When does every group with order divisible by $n$ have a subgroup of order $n$?
- Proving that a subgroup of a finitely generated abelian group is finitely generated
- For 3 numbers represented with $n$ bits in binary, how many bits is required for their product.
- $-1$ as the only negative prime.
- If a group is the union of two subgroups, is one subgroup the group itself?
- In $R$, $f=g \iff f(x)=g(x), \forall x \in R$
- Rabin and Shallit Algorithm
- How do you split a long exact sequence into short exact sequences?
- Hom functors and exactness

The universal mapping property of localizations (or fraction fields) yields an easy test for isomorphism, see the Corollary below, from Atiyah & MacDonald, *Commutative Algebra*, p. 39.

In your case $\,A\,$ is a domain and $\,S\,$ is the set of nonzero elements in $A.\,$ Since $\,S\,$ contains no zero-divisors, condition $(ii)$ in the Corollary simplifies to $\,g:A\to Q\,$ is an injection. So $\,B = Q\,$ is isomorphic to the quotient field of $A\,$ if $\,Q\,$ contains an isomorphic image $\bar A$ of $A$ such every nonzero $\,a\in A\,$ maps to a unit $\, \bar a = g(a)\,$ in $Q,\,$ and every $\,q\in Q\,$ is a fraction over $\bar A,\,$ i.e. $\,q = \bar a/\bar b = \bar a \bar b^{-1}\,$ for some $\,0\neq b,\,a\in A.$

- Prove the map has a fixed point
- $G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
- Find intersection of two 3D lines
- Examples of metric spaces which are not normed linear spaces?
- Foundation for analysis without axiom of choice?
- Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?
- Question Concerning Vectors
- How to write an integral as a limit?
- Distance between a point and a closed set in metric space
- Chain rule for discrete/finite calculus
- Intuition explanation of taylor expansion?
- Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?
- $\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$
- partitions with even number of even parts – partitions with odd number of even parts
- Sequential Criterion for Functional Limits