# Tagged Questions

**0**

votes

**2**answers

162 views

### power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...

**3**

votes

**1**answer

471 views

### When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...

**4**

votes

**2**answers

563 views

### Solving z^n=a+ib using only radicals of positive real numbers

Let $a+bi\in\mathbf{C}$ be a complex number with $a,b\in\mathbf{R}$. Then
it is easy to find exact solutions of $z^2=a+ib$. For example let $z=u+iv$. Then
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...

**2**

votes

**1**answer

271 views

### nth-powers and degree n polynomials with coefficients in field extensions

Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks

**3**

votes

**2**answers

324 views

### Rank of sum of Galois conjugates of a matrix

Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in ...

**4**

votes

**2**answers

625 views

### Splitting of a division algebra with an involution of second kind

Let $k$ be a field, $K/k$ a separable quadratic extension,
and $D/K$ a central division algebra of dimension $r^2$ over $K$
with an involution $\sigma$ of second kind
(i.e. $\sigma$ acts non-trivially ...

**30**

votes

**3**answers

4k views

### transcendental Galois theory

Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...