7
votes
1answer
72 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
1
vote
1answer
35 views

Prove that any two bases of a field extension have the same cardinality.

Suppose $E$ and $F$ are subfields of $\mathbb{R}$ with $F\subseteq E$. Prove that any two bases of $E/F$ have the same cardinality. The definition of a basis I am using is any finite set $S\subseteq ...
2
votes
1answer
100 views

Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
1
vote
3answers
136 views

Show that these matrices are congruent.

Let $K$ be a field of characteristic$\ne 2$ and $u$ be an invertible element of $K$. Show that $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\begin{pmatrix}u&0\\0&-u\end{pmatrix}$ are ...
1
vote
2answers
30 views

Show the following subspaces are invariant

Let $V$ be a vector space over a field $F$ and let $\alpha \in End(V)$. IF $W$ and $Y$ are subspaces of $V$ which are invariant under $\alpha$, show that both $W+Y$ and $W\cap Y$ are invariant under ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
0
votes
3answers
33 views

characteristic of a field>0

I have no idea how to solve such an exercise: Given is a field K, $ n \in \mathbb{N}, 1_K:=(1,1,..1) \in K^n $ $e_1,..,e_n$ is a standard basis of $K^n$. One show that: $1_K-e_1,..,1_K-e_n$ are ...
2
votes
3answers
39 views

Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
0
votes
1answer
27 views

Help Understanding Field Extension/Linear Algebra Problem

Let $E = F(\alpha)$ be a simple field extension of a finite field F by an algebraic element $\alpha$. Thinking of $E$ as an $F$-vector space, define a linear transformation $$T:E\rightarrow ...
3
votes
2answers
88 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
1
vote
5answers
86 views

Proving a structure is a field?

Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is: Let $F$ be a field and let $G=F\times F$. Define operations of addition ...
1
vote
1answer
65 views

Cyclotomic Cosets and Minimal Polynomial for 45

Currently I am working on matlab in order to find Cyclotomic Cosets for 45. As 45 in not in the format of 2^m-1, matlab give me an error. I am trying to write algorithm in matlab/octave for my ...
2
votes
2answers
32 views

A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in ...
0
votes
1answer
41 views

Why does the characteristic need to be 3?

and this is the solution given Why do we need the characteristic to be 3? Why wouldn't this work if over $\mathbb{Z}/\mathbb{9Z}$?
1
vote
2answers
38 views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & ...
2
votes
1answer
63 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
2
votes
0answers
31 views

Number of elements and number of different basis of $\mathbb F_5^3$

Let $\mathbb F:=\mathbb F_5$ the field with five elements. (i) How many elements has $\mathbb F^3$? (ii) How many different basis has $\mathbb F^3$? My idea: (i) $\mathbb F^3$ has $5^3$ elements. ...
0
votes
0answers
44 views

Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
0
votes
3answers
34 views

Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations have 1 solution.

Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations, say $7x-y=1$ and $11x+7y=3$ have 1 solution. I can give some values of $p$ like the obvious $p = 7, 11$. But how do I ...
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 ...
0
votes
1answer
39 views

Can I rotate divergence away? / Can get divergence from a rotation?

Let $v(x,y)$ be a two-dimensional vector field and let $R(x,y, \theta)$ be the two-dimensional rotation matrix which rotations a vector field around $(x,y)$ an angle $\theta$. The following two ...
4
votes
1answer
49 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
2
votes
3answers
68 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
1
vote
1answer
46 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
0
votes
1answer
75 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
2
votes
1answer
43 views

How to find all the roots in this ring?

Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$ Is it a field? Find all the roots in F of the polynom $f (Y) := Y^2+[3]_{F}Y +[1]_{F} \in F[Y]$. Attempt: It is a field, because $x^2+3x+1$ is irreducible ...
1
vote
0answers
40 views

Direct proof of: $\#($cover of $V) < \#F \; \Rightarrow \;V$ belongs to cover

I'm looking for a direct proof 1 (as opposed to a proof-by-contradiction) of the following theorem: Let $V$ be a vector space over a field $F$ and let $\mathcal{W}$ be a collection of (vector) ...
1
vote
1answer
59 views

Set of Functions is a Vector Space problem

Let $F$ be a field. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Define $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
21
votes
1answer
267 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
0
votes
2answers
94 views

Linear map on a finite dimensional vector space over an algebraically closed field of characteristic $p>0$

Let $V$ be a finite dimensional vector space over an algebraically closed field $F$ of finite characteristic $p$. Let $\alpha: V\longrightarrow V$ be a linear operator on $V$, and suppose that there ...
1
vote
0answers
37 views

integral domain with a field as a subring [duplicate]

I would like to know if my solution to the following exercise is correct. Let $A$ be an integral domain (with a unit) which has a field $\mathbb K$ as a subring and such that $A$ is a ...
3
votes
2answers
48 views

Basis elements for $Q(t)$ as a $Q(t^2)$-vector space.

Let $\mathbb{Q}(t)$ and $\mathbb{Q}(t^2)$ be the fields of rational functions with $t$ and $t^2$ as indeterminates. Both of these fields are infinite-dimensional. How can I determine the dimension of ...
3
votes
1answer
87 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
2
votes
1answer
120 views

Find all solutions for a system of linear equations over a given field

My Problem is: to find all Solutions for the following given System of linear equations over the Field $K = \mathbb{Z}_{/7}$ The System is given with: $$\begin{equation} \begin{split} ...
0
votes
1answer
32 views

Solving a linear equation in an extension field

Let $\mathbb K$ a commutative field and $\mathbb L$ an extension field of $\mathbb K$ (that is to say a field that contains $\mathbb K$). Let $A$ be a $n \times p$ matrix over $\mathbb K$. Let $B$ ...
2
votes
1answer
69 views

Vector space isomorphic fields

$\Bbb R(X)\simeq\Bbb R(X^2)$ but $[\Bbb R(X):\Bbb R(X^2)]\neq [\Bbb R(X):\Bbb R(X)]$. Is this correct? I thought the dimension of a vector space remains the same if I replace the field by an ...
1
vote
1answer
258 views

Find the degree and a basis for $\mathbb{Q}( \sqrt2, \sqrt3)$ over $ \mathbb{Q}( \sqrt2 +\sqrt 3)$

Similar questions were asked here before but I still can't find exactly what I want. I want to find the degree and a basis for $\mathbb{Q}( \sqrt2, \sqrt3)$ over $ \mathbb{Q}( \sqrt2 +\sqrt 3)$ I've ...
0
votes
1answer
37 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
3
votes
2answers
103 views

Tensor product of a field with itself.

I am proving the fact that if $A$ and $B$ are two central $k$-algebras where $k$ is a field (so then $Z(A) = Z(B) = k$), then $A \otimes B$ is also central. I made almost everything except this: I ...
2
votes
1answer
37 views

Equations over fields

Let $x_1,\cdots, x_n$ be distinct elements of a given field $F$ such that for any $k$, $\sum_{i=1}^n x_{i}^k = 0$. I want to show that all $x_i$'s are zero.
0
votes
1answer
66 views

Why is (gcd(f,g)) = (f,g)?

f and g are polynomials of F[X]. I can't see why (f,g) = (gcd(f,g)) ? (the ideal that f and g are the generators, and the ideal that the gcd is the generator). gcd(f,g) = a*f+b*g , for specific a ...
0
votes
1answer
70 views

Find the multiplicative inverse of $\,x^2+(x^3-x+2)$ in the quotient $\,F_3[x]/(x^3-x+2)$

Find the multiplicative inverse of $x^2+(x^3-x+2)$ in the quotient $F_3[x]/(x^3-x+2)$ . I've proved that $x^3-x+2$ is irreducible polynomial in $F_3[x]$, and that $x^2$ and $x^3-x+2$ are coprime ...
1
vote
1answer
63 views

On the extension of fields

Let $F\subseteq K$ be a finite field extension and let $a_1,..., a_n$ be an $F$-basis for $K$. I want to show that the matrix $A := (tr(a_ia_j))$ is singular if and only if $tr K =0$. Any suggestion ...
2
votes
2answers
55 views

Is there an axiomatic definition of the concept “field equipped with a conjugation operator”?

In some sense, $\mathbb{C}$ is more than just a field, since aside from the usual field operations, it is also equipped with a conjugation operator $\mathbb{C} \rightarrow \mathbb{C}$. This means a ...
1
vote
1answer
41 views

Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
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 ...
1
vote
1answer
551 views

Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself? Intuitively, it seems obvious because the definition of a field is nearly the same as that of a vector space, just with scalers instead of ...
1
vote
2answers
87 views

Significance of the trace in isomorphic matrix fields

The field $\mathbb{Q}(\operatorname{i})$ has an isomorphic matrix field of degree two. The isomorphism is $$\varphi:x+\operatorname{i}\!y \longmapsto \left[\begin{array}{cc} x & -y \\ y & x ...
6
votes
1answer
126 views

Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
2
votes
1answer
174 views

Understanding Dummit & Foote p.528 Sec 13.2 Algebraic Extensions

I can't understand why the authors conclude Hence, the elements $\alpha_i\beta_j$ span the composite extension $K_1K_2$ over $F$. I would like to understand what the authors mean and how they ...