# Tagged Questions

**4**

votes

**1**answer

396 views

### Examples of “nice” properties of algebraic extensions of $\mathbb{Q}$

I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if
every finite extension of $\mathbb{Q}$ satisfies (P), and
if $K ...

**23**

votes

**3**answers

2k views

### Arithmetic geometry examples

(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...

**4**

votes

**2**answers

3k views

### Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...

**21**

votes

**2**answers

2k views

### Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...

**13**

votes

**3**answers

1k views

### Eta-products and modular elliptic curves

Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form
$$
...

**5**

votes

**2**answers

1k views

### Example of connected-etale sequence for group schemes over a Henselian field?

Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...

**44**

votes

**2**answers

4k views

### Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...

**39**

votes

**28**answers

18k views

### Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant GĂ¶del ...

**18**

votes

**11**answers

3k views

### When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...

**14**

votes

**5**answers

2k views

### Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, ...