# Tagged Questions

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 ...
42 views

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 $\prod_{\sigma ... 1answer 74 views ### Having trouble considering a finite field$\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$. How can we work with this vector space? ... 2answers 154 views ### Characteristic Polynomial of Galois automorphism Let$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$? 0answers 44 views ### diagonal action induces permutation Suppose one has two$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 $$(c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)})$$ ... 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 ... 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 ... 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 ... 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 ...
$\def\Fp{\mathbb F_p}$ 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 ...
### 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 ...