2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
0
votes
1answer
16 views

Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, ...
2
votes
2answers
53 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
4
votes
2answers
268 views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
6
votes
2answers
277 views

Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
2
votes
1answer
53 views

Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
12
votes
0answers
108 views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
3
votes
0answers
59 views

$\mathbb{Z}[ \sqrt{−n}]$ is not PID [duplicate]

a) I prove that for $n=5$,$n=3$ that $\mathbb{Z}[ \sqrt{−n}]$ we have that 2 is irreducible but not prime, but how can i prove that in general for $n \geq 3$ $\mathbb{Z}[ \sqrt{−n}]$ 2 is irreducible ...
0
votes
5answers
375 views

Is $x^8+1$ irreducible in $\mathbb{R}[x]$

Question is to check if : $x^8+1$ is irreducible over $\mathbb{R}[x]$. even before this I tried to see $x^4+1$ and $x^2+1$. for $x^2+1$, it does not have a root in $\mathbb{R}$ So, it is ...
1
vote
1answer
72 views

proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions

Let $A$ be an integrally closed integral domain with field of fractions $K$ and let $p \in Spec(A)$. Let $L/K$ be a Galois extension with group $G$ and $L'/K$ be a finite Galois subextension. Let $B$ ...
2
votes
1answer
79 views

Compute $\mathrm{Aut}(S)$ of the ring $S=\mathbb{Q}[x]/(x^2)$

This problem once again is from a previous exam. The problem is to compute the group of automorphisms of the ring $S$ where $S=\mathbb{Q}[x]/(x^2)$ My thoughts: Well $x^2$ is reducible over ...
7
votes
0answers
78 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
2
votes
2answers
136 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
4
votes
2answers
144 views

Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to ...
3
votes
1answer
169 views

Counting endomorphisms of $\mathbf Q(\zeta _{n})$

If $\zeta= \zeta_{n}$, how does one count the homomorphisms $f:\mathbf{Q}(\zeta)\rightarrow \mathbf{Q}(\zeta)$?
2
votes
1answer
70 views

For $n\ge 3, x_{1},…,x_{n} \in \mathbf{Q}^{\ast}$, $[\mathbf{Q}(\sqrt{x_{1}},…\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$

For $n\ge 3, x_{1},...,x_{n} \in \mathbf{Q}^{\ast}$ and $[\mathbf{Q}(\sqrt{x_{1}},...\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$ how can we conclude that there are non empty $I \subset \{1,...,n\}$ with ...
9
votes
3answers
353 views

Galois ring extension

Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.