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**8**

votes

**2**answers

418 views

### Evidence for Q^solv being Pseudo-algebraically-closed

This is a follow-up to the following answer:
Solvable class field theory
in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed ...

**2**

votes

**4**answers

501 views

### a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...

**10**

votes

**1**answer

565 views

### a question on function fields

Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...

**27**

votes

**6**answers

4k views

### Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...

**8**

votes

**2**answers

1k views

### Characterisation for separable extension of a field

Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L ...

**15**

votes

**6**answers

2k views

### In what sense are fields an algebraic theory?

Since there is no "free field generated by a set", it would seem that
1) there is no monad on Set whose algebras are exactly the fields
and
2) there is no Lawvere theory whose models in Set are ...

**44**

votes

**9**answers

9k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**-3**

votes

**1**answer

590 views

### Fundamental: Division by Zero [closed]

All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions.
...

**16**

votes

**4**answers

5k views

### Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...