# Tagged Questions

**17**

votes

**1**answer

952 views

### How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...

**4**

votes

**0**answers

133 views

### The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...

**5**

votes

**2**answers

406 views

### Which polynomials are Fricke polynomials ?

Let me recall the definition which seems the most standard of Fricke polynomials.
Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique ...

**1**

vote

**2**answers

170 views

### Completing unimodular vectors with $3$ entries in $F_2[t]$ to a $3$ by $3$ matrix with determinant equal to $1.$

Given $a_1,b_1,c_1$ in $F_2[t]$ with $\gcd(a_1,b_1,c_1)=1$ it is known that there exists an element
$g$ of $SL(3, F_2[t])$ (by explicit construction) such that $g$ has first line
$$
[a_1^2, b_1,c_1],
...

**3**

votes

**2**answers

335 views

### Polynomial group Laws on $\mathbb{R}^2$

When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation ...

**6**

votes

**0**answers

198 views

### Involution on sextic polynomials?

The strangeness of $Aut(S_{6})$ suggests the following question:
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be the zeros of a 6th degree polynomial $P$ over a field $F$.
Let ...

**4**

votes

**2**answers

1k views

### Algebraic integers on the unit circle

Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. Do these objects have a name?
I would guess they contain useful ...

**1**

vote

**3**answers

446 views

### A question regarding polynomials whose roots satisfy certain algebraic relation

Suppose I know the following information about a function :
1) Its a polynomial (not an explicit equation, neither the roots nor the degree is known)
2) I have managed to find an algebraic relation ...

**9**

votes

**2**answers

1k views

### How to show the galois group of a polynomial is not an alternating group?

I am trying to prove that a certain class of polynomials have symmetric galois group.
Using the Newton polygon, I have shown that the galois groups of these polynomials are transitive on $k$-sets for ...