1
vote
2answers
46 views

Trace/Norm of Field Extension vs Trace/Determinant of Linear Operators

Dummit and Foote (3rd ed, page 582-3) defines the norm and trace of an element of a field extension as follows: Let $K/F$ be any finite field extension, and let $\alpha\in K$. Let $L$ be a Galois ...
2
votes
1answer
42 views

Charactristic polynomial of a F-linear transformation with respect to Galois group

Let $K$ be a Galois extension of $F$, and let $a \in K$. Let $L_a : K \to K$ be the $F$-linear transformation defined by $L_a(b)=ab$. Show that the characteristic polynomial of $L_a$ is $\prod_{\sigma ...
2
votes
1answer
74 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
3
votes
2answers
154 views

Characteristic Polynomial of Galois automorphism

Let $K/F$ be a finite Galois extension. Let $g$ be an element of $Gal(K/F)$ How do I compute the characteristic polynomial of $g$, where $g$ is considered as a $F$-linear map $K \rightarrow K$?
2
votes
0answers
44 views

diagonal action induces permutation

Suppose one has two $n$-tuples of complex numbers $(c_1,\dots,c_n)$ and $(z_1,\dots,z_n)$ such that all $c_i$, $z_i$ are nonzero, and $$ (c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)}) $$ ...
0
votes
1answer
43 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
2
votes
1answer
107 views

Field not closed under complex conjugation

What is an example of an algebraic field not closed under complex conjugation? In all subfields of $\mathbb C$ I think of, complex conjugation is a transposition. I think I understand that it is ...
5
votes
2answers
118 views

Questioning a Basis for $\mathbb{Q}[\sqrt[3]{2}]$ over $\mathbb{Q}$

Let $\omega = e^{2 \pi i /3}$ and $\alpha = \sqrt[3]{2}$. I'm seeing it claimed that $\mathcal{B} = \{\alpha, \alpha^2, \omega \alpha, \omega \alpha^2, \omega^2 \alpha, \omega^2 \alpha^2\}$ forms a ...
4
votes
1answer
67 views

Vector Space isomorphisms of $\mathbb{Q}(z)$ preserving the Galois group (where $z$ is a primitive third root of unity)

Take the field extension $\mathbb{Q}(z)$ where $z$ is a primitive third root of unity and consider the set $A$ of vector-space automorphisms of $\mathbb{Q}(z)$ so that for $T \in A$ the map $\phi ...
-5
votes
1answer
247 views

In Field Theory, some basic problems

$\def\Fp{\mathbb F_p}$ 1. Determine whether the following statements are True of False. Give brief reasons. (A) Let $u$ and $v$ be indeterminates. The field $\Fp(u,v)$ has a primitive element over ...
2
votes
0answers
82 views

Given a normal basis $\{g(a)| g\in Gal(L/K)\}$ of $L/K$ finite Galois, can we express $g(a)g'(a)$ relative to this basis?

Let $L/K$ be a finite Galois extension. The normal basis theorem says that there is an element $a\in L$ such that $\{ g(a) | g\in \text{Gal}(L/K)\}$ is a basis of $L$ as a $K$-vector space. Let ...