Intereting Posts

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Green's function of a bounded domain is strictly negative
Does the Levi-Civita connection determine the metric?
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Can RUBIK's cube be solved using group theory?
Getting equation from differential equations
Detecting that a fraction is a repeating decimal
Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?
Compactly supported function whose Fourier transform decays exponentially?
Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”
Confused about dimension of circle
Enumerating Graphs with Self-Loops
How to solve this Initial boundary value PDE problem?
What's wrong with my solution for the birthday problem?

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields?

My guess is yes, because every morphism of fields is injective, hence in this case $B$ contains an isomorphic copy of $A$, which in turns contains one copy of $B$. If this is right, how can i formalize it?

- Projectivity of $B$ over $C$, given $A \subset C \subset B$
- When do we use Tensor?
- Step in Proof of Cardinality of Product of two Groups
- Characteristic of a Non-unital Integral Ring
- What is the field of fractions of $\mathbb{Q}/(x^2+y^2)$?
- Why do we have to do the same things to both sides of an equation?

- Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$
- “Direct sums of injective modules over Noetherian ring is injective” and its analogue
- If each component of a Cartesian product is homeomorphic to another space, are the Cartesian products homeomorphic
- Maximal ideal/ Prime ideal
- No group of order 400 is simple
- Every finite group is isomorphic to some Galois group for some finite normal extension of some field.
- If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
- Extending Homomorphism into Algebraically Closed Field
- A finite field cannot be an ordered field.
- If $a^m=b^m$ and $a^n=b^n$ for $(m,n)=1$, does $a=b$?

This is an occasion, when instincts developed over finite extensions of (prime) fields lead one astray.

The first counterexamples that come to mind need a bit of background from the theory of elliptic curves. It is quite possible for there to be isogenies going back and forth between two non-isomorphic elliptic curves, $E_1$ and $E_2$. The isogenies give rise to embeddings between the corresponding function fields

$K(E_1)$ and $K(E_2)$ (take for example $K=\mathbb{C}$ to avoid several algebraic pitfalls). Yet, if the two elliptic curves are not isomorphic, the functions fields won’t be isomorphic either.

- Number of monic irreducible polynomials of prime degree $p$ over finite fields
- Minimal generating sets of free modules, and endomorphisms of free modules
- What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…
- Intuition of the meaning of homology groups
- invertible if and only if bijective
- How find the sum of the last two digits of $(x^{2})^{2013} + \frac{1}{(x^{2})^{2013}}$ for $x + \frac{1}{x} = 3$?
- the sixth number system
- Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$
- Asymptotic behaviour of the length of a curve .
- Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$
- Smooth spectral decomposition of a matrix
- Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? And general tips about series and the logarithm
- If $G/Z(G)$ is cyclic, then $G$ is abelian
- Prove that the intersection of two equivalence relations is an equivalence relation.
- Integral equal $0$ for all $x$ implies $f=0$ a.e.