Tagged Questions

1
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1answer
33 views

Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$

For $n \geq 5$ prime number, calculate the sum of: $$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$ under $\mathbb{Z}_n$. I figured it's the hyperharmonic\over-harmonic series, $$ ...
3
votes
5answers
57 views

On any finite field, adding the identity element a finite amount of times will result to the neutral element

Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$). I started with saying that in every field there's an element $a \in F$ so that $a + ...
1
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2answers
37 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
0
votes
1answer
23 views

Why does the minimal polynomial not change on changing the field?

Suppose $V$ is a vector space over a field $K$, $F$ is a subfield of $K$ and $T:V(F)\to V(F)$ is a linear operator. I think that if the field $F$ is changed to $K$ the characteristic polynomial of $T$ ...
2
votes
1answer
42 views

Basis of Polynomials over a Finite Field

I'm trying to show that $1, x, x^2, .... x^k$ are a basis for the space of all polynomials with degree less than or k over a finite field $\mathbb{K}$ with p elements. I know that over the reals I ...
2
votes
2answers
50 views

Is $N_{k\subset K}$ the only *norm* on the field extension $k\subset K$?

In several examples of field extensions the norm function is very useful. For instance, in $\mathbb{Q}\subset\mathbb{Q}(\sqrt{2})$, the norm is $N(x+y\sqrt{2})=x^2-2y^2$. In ...
10
votes
1answer
81 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
1
vote
1answer
97 views

Linear independence in Finite Fields

How can we define linear independence in vectors over $\mathbb{F_{2^m}}$ ? Let vectors $v_1,v_2,v_3$ $\in$ $\mathbb{F_{2^m}}$, If $v_1,v_2,v_3$ are linearly independent,then ...
4
votes
2answers
161 views

Why are fields with characteristic 2 so pathological?

For example, over fields with characteristic 2, there exist nonzero symmetric nilpotent matrices, and nonzero matrices could be simultaneously symmetric and anti-symmetric. I wonder why characteristic ...
4
votes
1answer
145 views

Showing a polynomial not reducible.

How do I show that $f(x)=1+2x+\cdots+(p-1){x}^{p-2}$ is not reducible on $\mathbb{Q}$, where $p$ is prime.
1
vote
1answer
41 views

Using Smith Normal Form to prove that even for a singular square matrix with entries in field $F$, $\exists U$ nonsingular where $(UA)^2 = UA$

How might I use Smith Normal Form of a matrix to show that even for a singular square matrix with entries in field $F$, $\exists U$ nonsingular (with entries in $F$) where $(UA)^2 = UA$? The ...
1
vote
1answer
37 views

Dependence of $|K|$ and the union of the subspaces.

If $V$ is a $K$-vector space with $|K|\ge n$ and $V_1, \ldots, V_n$ are subspaces of $V$ such that $V=V_1 \cup \cdots \cup V_n$, then $V=V_i$ for some $i$. Is there a counterexample that for $|K|< ...
0
votes
1answer
56 views

Approximate rational number of radical combination

Suppose that there is a radical combination $a\sqrt{b}+c\sqrt{d}$ where a,b,c,d are natural. Each term part $\sqrt{b}$ cannot be transformed into the form of $s\sqrt{q}$. The question is, 1) Suppose ...
3
votes
1answer
89 views

Linear Transformation over Subfield

Letting $F\subseteq K$ be fields, and $V$ a vector space over $K$. Certainly, $V$ is also a vector space over $F$. And if $\{e_1,...,e_n\}$ is a basis for $K$ over $F$ and ...
0
votes
3answers
84 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
2
votes
2answers
69 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
2
votes
1answer
84 views

Field Extension of the Rationals

So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things: ...
-6
votes
1answer
188 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 $v$ are indeterminates. The field $\Fp(u,v)$ has a primitive element over ...
2
votes
0answers
58 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 ...
7
votes
1answer
163 views

If $K=K^2$ then every automorphism of $\mbox{Aut}_K V$, where $\dim V< \infty$, is the square of some endomorphism.

I have to show the following: Let $K$ be a field such that $\mbox{char } K \neq 2$ and each element of $K$ is a square (i.e. $K^2=K$) and let $V$ be a finite-dimensional vector spaces over $K$. ...
0
votes
1answer
79 views

What is a “p-linear endomorphism”?

I came across this terminology but don't know what it means: "p-linear endomorphism". More specifically, it said "Let F be a p-linear endomorphism of a vector space V". Can someone explain it to me? ...
1
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2answers
106 views

how do I prove that in a finite field, the exponent of a sum is the sum of the exponents?

I did some search over the website and in google, but couldn't find the answer, even though I'm sure it exists, so maybe I did not look for the right key words. Anyway, the question is how do I prove ...
2
votes
1answer
422 views

Finding a Basis for a Field Extension

I am asked to find the degree and basis for a given field extension $\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{6},\sqrt[3]{24}) $ Now I know that the degree for each vector is $3$, and that the basis will have ...
1
vote
1answer
46 views

Characterization of fields $\mathbb{F}$ with the property: there exists a vector space $V$ such that $V^{\star}$ is isomorphic to $\mathbb{F}[X]$

In this post, Georges Elencwajg says that Erdős-Kaplansky tell us that there does not exist a real vector space whose dual is isomorphic to R[X] ! I would like to know if there is a ...
2
votes
0answers
201 views

Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...
1
vote
3answers
65 views

Find members of this field.

Let $m$ be an integer, $m \geq 2$. and let ${\mathbb Z}_m$ be the set of all positive integers less than $m$, ${\mathbb Z}_m = \{0, 1, ..., m -1\}$. If $a$ and $b$ are in ${\mathbb Z}_m$, let $a + b$ ...
2
votes
1answer
75 views

number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group

My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group? I know that there are abelian groups that we can not endow ...
13
votes
2answers
604 views

Basis of primitive nth Roots in a Cyclotomic Extension?

While reading one of Keith Conrad's great blurbs, Linear Independence of Characters, there is a footnote at the bottom of page 2 saying In general, the primitive $n$th roots of unity in the $n$th ...
1
vote
3answers
226 views

Eigenvalues of a linear operator over a K-vector space

The following question may be somewhat ill-posed due to a lack of experience in dealing with fields. Let $T$ be a linear operator over a finite dimensional complex vector space. If the only (complex) ...
0
votes
1answer
143 views

Principle of substitution — fields

Could someone possibly explain this definition (applied to fields) to me? The principle of substitution: In a field F, we can, in any formula involving an element $\alpha\in F$, replace $\alpha$ ...
5
votes
2answers
157 views

How does extending a field affect matrix similitude?

Suppose we have fields $\mathbb{K}_1, \mathbb{K}_2$ s.t. $\mathbb{K}_1 \subset \mathbb{K}_2$. On a paper I'm reading there is a flashy reference to some algebraic results concerning similitude of ...
4
votes
3answers
422 views

What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...