Intereting Posts

Are injectivity and surjectivity dual?
Prove that any polynomial in $F$ can be written in a unique manner as a product of irreducible polynomials in F.
convergence of rather uncommon series
Finding the angle of rotation of an ellipse from its general equation and the other way around
Whether the code produced by CRC is a cyclic code?
Applications of algebra and/or topology to stochastic (or Markov) processes
What are some examples of classes that are not sets?
Evaluation of $\int_{0}^{1} \frac{dx}{1+\sqrt{x}}$ for $n\in\mathbb{N}$
How many ways there are?
Example of a normal extension.
Integrate Form $du / (a^2 + u^2)^{3/2}$
Show $1/(1+ x^2)$ is uniformly continuous on $\Bbb R$.
Simple question about the definition of Brownian motion
Probability of cars being blocked during red light
Colimit of $\frac{1}{n} \mathbb{Z}$

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property:

every maximal ideal of this ring is the subset of all functions vanishing at a common point.

If we follow the argument in the proof of this property, it can be seen that $\mathbb{R}$ can be replaced by $\mathbb{C}$ and still similar property holds for new ring (we have to slightly modify argument).

- On $GL_2(\mathbb F_3)$
- Sufficient conditions for being a PID
- Is there a general formula for finding all subgroups of dihedral groups?
- Minimal polynomial of $\sqrt{5}+\sqrt 2$ over $\mathbb{Q}(\sqrt{5})$
- Is $\mathbb{Z}/3\otimes \mathbb{Z}=\mathbb{Z}/3$?
- Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

What if we place $\mathbb{Q}$? Here $\mathbb{Q}$ is disconnected with induced topology from $\mathbb{C}$ and so the only continuous function from $[0,1]$ to $\mathbb{Q}$ are constant (am I correct?). So, the property no longer holds true with replacement of $\mathbb{R}$ by $\mathbb{Q}$.

**Question:** Are there subfields of $\mathbb{C}$ other than $\mathbb{R}$ which are connected as topological spaces with induced topology from $\mathbb{C}$?

- Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.
- What is a projective ideal?
- Adjoining a number to a field
- Oh Times, $\otimes$ in linear algebra and tensors
- Why do direct limits preserve exactness?
- Modular Forms: Find a set of representatives for the cusps of $\Gamma_0(4)$
- How can we show that an abelian group of order <1024 has a set of generators of cardinality <10
- Finite ring of sets
- Connections between number theory and abstract algebra.
- Are there any theories being developed which study structures with many operations and many distributive laws?

Yes, there are proper connected subfields of $\mathbb{C}$ other than $\mathbb{R}$.

A construction can be found in a paper by J.Dieudonné:

J.Dieudonné, *Sur les corps topologiques connexes*, C.R. Acad. Sci. Paris vol. 221 (1945) pp. 396-398.

I don’t have access to the above paper at the moment.

It can in fact be proved that the only proper path-connected subfield of $\mathbb{C}$ is $\mathbb{R}$, see a more detailed discussion at:

Niel Shell, *Connected and disconnected fields*, Topology and its Applications, (27) 1987, 37-50.

Also, since the only maximal ideal of $\mathbb{Q}$ is $\{0\}$, the statement about maximal ideals remains true.

- Fermat's Last Theorem simple proof
- Is the function $F(x,y)=1−e^{−xy}$ $0 ≤ x$, $y < ∞$, the joint cumulative distribution function of some pair of random variables?
- Any good approximation for this integral?
- Injective, surjective and bijective for linear maps
- Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?
- Complex integral help involving $\sin^{2n}(x)$
- Solving $\lim\limits_{x\to0} \frac{x – \sin(x)}{x^2}$ without L'Hospital's Rule.
- Homology and Graph Theory
- Infinitely times differentiable function
- Regular monomorphisms of commutative rings
- Reasons for coherence for bi/monoidal categories
- Ramsey Number Inequality: $R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$
- Field axioms: Why do we have $ 1 \neq 0$?
- Prove $A = (A \setminus B) \cup (A \cap B)$
- Homology of a simple chain complex