Tagged Questions
1
vote
1answer
24 views
If $F$ is a finite field of size $q=p^n$ and $b\in K$ is algebraic over $F$ then $b^{q^{m}} = b$ for some $m > 0$
I want to apply a similar type of argument to show that $\alpha^{q} = \alpha$ for all $\alpha\in F$. When we know that the characteristic of $F$ is $p$ and $|F| = q = p^{n}$. But I dont know how to ...
2
votes
2answers
33 views
Find $u\in\mathbb{R}$ such that $\mathbb{Q}(u) = \mathbb{Q}(2^{1/2}, 5^{1/3})$.
I am having trouble finding such a $u$. My instincts at first told me to do the obvious thing and let $u = 2^{1/2}5^{1/3}$ but $u^{2} = \left(2^{1/2}5^{1/3}\right)^{2} = 2\cdot5^{2/3}$ but we want ...
0
votes
0answers
55 views
Automorphisms of fields
How can I prove that there is an element of order 23 in $\mathrm{Aut}\mathbb{Q}(K)$, where K is a subfield of complex numbers generated by all complex roots of $x^{23}-6x^{22}+3$?
2
votes
1answer
30 views
Finite Extensions and Roots of Unity
Two questions; the hint I've been provided is that they are, in fact, related.
Prove that a finite extension of $\mathbb{Q}$ contains finitely many roots of unity.
What is the largest (finite) ...
2
votes
2answers
17 views
Relationships of Eigenvalues in Algebraic Closure
Suppose that $k$ is a field, and $A \in M_n(k)$ is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not ...
2
votes
0answers
30 views
Non-isomorphic simple extensions of the same degree of a field of positive characteristic
Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic.
I thought of an example where they are ...
1
vote
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 + ...
0
votes
2answers
37 views
Addition table for a 4 elements field
Why is this addition table good,
\begin{matrix}
\boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\
\boldsymbol{\textbf{}0} & 0 & 1 ...
1
vote
1answer
50 views
Field extension of $\mathbb Q$ of degree 2
Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$.
I know that $K$ is generated by say ...
1
vote
1answer
65 views
Characterization of transcendental elements in algebraic function fields
I would like to prove this equivalence:
Let $F|K$ be an algebraic function field. Then $z \in F$ is transcendental over $K$ if and only if $[F:K(z)] < \infty$. (This statement is Remark 1.1.2 ...
1
vote
0answers
23 views
Let $F$ be a field and $E=F(a)$, where the min poly of a has degree n over F. What are the conditions on $m\in \mathbb{Z}$ so that $E=F(a)=F(a^m)$
So far I've been playing around with different techniques. I tried to show double inclusion to get the equality in hopes of coming across a needed condition on m, but was unsuccessull. I also worked ...
4
votes
1answer
43 views
No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$
Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
2
votes
2answers
75 views
Proving that the inverse of an algebraic element is algebraic
Let $E$ be an extension of a field $F$. Suppose $c \in E$ is algebraic over $F$ where $c \neq 0$. I want to prove that $c^{-1}$ is also algebraic over $F$.
I feel like I'm missing something obvious ...
0
votes
2answers
85 views
Number of irreducible polynomials of degree $3$ over $\mathbb{F}_3$ and $\mathbb{F}_5$.
I'm trying to find the number of $3$rd degree irreducible polynomials over $\mathbb{F}_3$ and $\mathbb{F}_5$.
Since a $3$rd degree polynomial is irreducible if and only if it is divisible by a ...
4
votes
2answers
76 views
Radical extension
Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$.
It's ...
4
votes
1answer
57 views
field of algebraic numbers
Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals?
Many Thanks
2
votes
0answers
71 views
Fixed points of automorphism in the field $\mathbb{C}(x,y)$
I am trying to solve a problem, and one of the parts is the following:
let $M=\bigl(\begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)$ be a non singular $2\times 2$ matrix with integer ...
2
votes
2answers
71 views
Isomorphisms between $\mathbb C$ and field $\mathbb K$
There is a field $\mathbb K$.
I've got an injective homomorphism $\varphi: \mathbb R \rightarrow \mathbb K$. Also I got $i \in \mathbb K$ with $i\cdot i = -1$.
I have to show, that there are ...
4
votes
1answer
50 views
Find separable irreducible $g$ such that $f(x)=g(x^{p^d})$
This is an exercise from VII.4. in Algebra: Chapter 0.
Let $\mathcal{k}$ be a field of characteristic $p$, and $f(x)\in\mathcal{k}[x]$ an inseparable irreducible polynomial. Find a separable ...
6
votes
3answers
199 views
How to show that a finite comutative ring without zero divisors is a field?
$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field?
I'm just starting with abstract algebra and I'd really appreciate if someone ...
2
votes
1answer
88 views
Number of monomorphisms $\mathbb{Q} \to \mathbb{C}$
In my abstract algebra class, we have been tasked to find all monomorphisms $\mathbb{Q} \to \mathbb{C}$. The book (Stewart's Galois Theory) gives an example for $\mathbb{Q}(\sqrt[3]{2}) \to ...
3
votes
1answer
139 views
Calculate $\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})$
I'm attempting some of my first problems in solving for Galois Groups, and this one has stumped me.
What I've done so far is found that $\mathbb{Q}(\sqrt[5]{3})$ is not a normal extension, because ...
5
votes
1answer
141 views
Calculating Splitting Field Degree of Extension
Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
2
votes
1answer
75 views
Countable Field Extension of a Countable Field
Okay, first question on this site, apologies in advance for any mistakes I may make.
Question:
So I need to show that an algebraic field extension $E:F$, with $F$ being countable, is countable.
My ...
1
vote
1answer
124 views
Finite Subgroups of Multiplicative Group of Field
Question:
Let F be a field of characteristic $0$ such that $|F:\mathbb Q|=2$, and let U be a finite subgroup of F*, the multiplicative group of F. Show that $|U|$ is 1, 2, 3, 4 or 6.
Attempt at ...
3
votes
4answers
130 views
Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
I need to rationalize $\displaystyle\frac{1}{a+b\sqrt[3]2 + c(\sqrt[3]2)^2}$
I'm given what I need to rationalize it, namely ...
0
votes
2answers
62 views
Inverses of elements in field extensions and inverses of basis elements
Say I create an extension $K$ over a field $F$ obtained by adjoining an element $\alpha$, i.e. $K = F[\alpha]$ ($\alpha$ does not necessarily have to be the root of a polynomial with coefficients in ...
3
votes
1answer
101 views
Splitting field of $x^{{p}^e}-1$ over $\mathbb Z_p$
I'd like a hint for determining the splitting field of $x^{{p}^e}-1$ over the integers mod $p$, $\mathbb Z_p$, where $e$ is an arbitrary natural number. Thanks.
4
votes
3answers
115 views
Showing that $\mathbb{F}_p(x)$ is an infinite field of finite characteristic
Let $F := \mathbb{F}_p(x)$, the field of rational functions in one variable over the prime field $\mathbb{F}_p$. How can we show that $F$ is an infinite field of finite characteristic?
Thoughts so ...
3
votes
5answers
266 views
Isomorphism between $\Bbb Q(i)$ and $\Bbb Q(\sqrt2)$
How can I show that $\Bbb Q(i)$ and $\Bbb Q(\sqrt2)$ as fields are not isomorphic. Is there an element of order $4$ in $\Bbb Q(\sqrt2)$?
3
votes
1answer
134 views
Splitting fields and Galois extensions
Let $L/K$ be a field extension such that $L$ is a splitting field of $f\in K[X]$, i.e. $f=\prod_{i=1}^{k} (X-u_i)^{n_i}$ for some $u_i\in L$. If we denote the coefficients of $g:=\prod_{i=1}^{k} ...
2
votes
1answer
45 views
Show a polynomial splits in an extension
Consider the following:
Let $E/K$ be a separable field extension of degree $p$ ($p$ a prime). Suppose $f\in K[x]$ is an irreducbile polynomial which has more than one root in $E$. Show $f$ splits ...
2
votes
3answers
142 views
$\mathbb{R}\subset\mathbb{F}\subset\mathbb{C}$ implies $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$?
Let $\mathbb{F}$ a field such that $\mathbb{R}\subset\mathbb{F}\subset\mathbb{C}$. Prove that $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$.
What I know is, that if ...
0
votes
2answers
77 views
How to prove that for every finite field its cardinality is $p^n$?
How to prove that for every finite field its cardinality is $p^n$ where $p$ is prime and $n\in\mathbb{N}$?
Thanks in advance!
1
vote
1answer
59 views
Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ with the following property?
Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ (proper subset) as follows:
$\mathbb{R}$ is a vector space over $K$ and has no finite generating set and $K$ is a vector ...
0
votes
3answers
69 views
Technique for showing an element is not in a field?
I have an extension $\mathbb{Q}(5^{1/4}, i)$, and I want to show that $4^{1/4}$ is not contained in it.
(I hope what I am trying to prove is true!)
Anyways, my natural starting point is to assume ...
1
vote
1answer
46 views
If $F\subseteq R \subseteq E$ where $E$ is an extension of $F$ and $R$ is an $F$-subspace, show that $R$ is a field.
I have another abstract algebra question. I stated it in the title, but here it is in more detail:
Let $F\subseteq R \subseteq E$ where $E$ is an algebraic extension of the field $F$. If $R$ is an ...
2
votes
1answer
56 views
Construction of a field
Given the polynomial
$$f(x)= x^4-16x^2+4$$
which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal ...
0
votes
1answer
47 views
Endomorphism of Algebraic Closure
This is a homework problem and I have some questions regarding it (not looking for solution):
Let $\tau$ be an endomorphism of a field $F$ that is algebraically closed.
i.e. $\tau: F\rightarrow ...
0
votes
2answers
82 views
Normality and Field Extensions
Let M : L : K be finite field extensions. When M is not normal over K, give four
examples to show that this gives no information about the normality of M over L or
of L over K. What are the ...
7
votes
2answers
217 views
In field ($F, +, \cdot$) , how can I prove $x^2 =1\implies x=1,-1$
I'm a really confused about fields.
I know that it means $x$ is the reciprocal element of itself, and I can easily show that $1^2=1$ (not as trivial for $(-1)^2$ though), but I'm not sure how it ...
4
votes
1answer
141 views
An angle $\theta$ can be trisected if and only if $4x^3-3x+\cos\theta$ is reducible over $\mathbb{Q}(\cos\theta)$
I was given the above problem for homework. There is (what seems to be) a relevant proof in my textbook regarding the impossibility of trisecting $\pi/3$. In this proof, the identity
$$\cos 3\theta = ...
1
vote
1answer
147 views
Non-trivial finite purely inseparable extension
QUESTION: Suppose K/F is a non-trivial finite degree purely inseparable extension.
Prove that there is a purely inseparable degree p extension of K.
I know, or can prove, that [K:F] is a power of p ...
1
vote
1answer
85 views
Field theory, extensions
HiAll, I am stuck with this problem:
(a) Let $K$ be a field such that characteristic of $K$ is not 2. Prove that any extension $L$ of $K$ with $K\subset L$, and $[L:K]=2$ has the form $L=F(\beta)$ ...
4
votes
1answer
111 views
Showing existence of a field extension of degree $n$ for a finite field $F$
EDIT: Just mentioning that this is a homework question.
This is my first time posting a question on math.stackexchange, so I hope you find it in your hearts to forgive any stylistic or rule ...
1
vote
2answers
93 views
$K$ finite extension of $F$ s.t. for every 2 subextensions $M_1, M_2$, $M_1\subset M_2$ or $M_2\subset M_1$. Then there's $a\in K$ such that $K=F(a)$
Let K be a finite extension of a field F such that for every two intermediate field $M_1$, $M_2$ we have $M_1\subset M_2$ or $M_2\subset M_1$. I need to show that there is an element $a\in K$ such ...
0
votes
1answer
38 views
probability inequality resolution finite field
I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a ...
1
vote
3answers
57 views
Multiplication in the field $F = \mathbb{Z}_2[x]/f(x)$
Let $f(x) = x^6 + x + 1$ and define the field $F = \mathbb{Z}_2[x]/f(x)$
Compute the following in this field:
1. $(x^5 + x + 1)(x^3 + x^2 +1)$
I start by multiplying (in $\mathbb{Z}_2[x]$):
...
1
vote
1answer
125 views
Quick way to check if a polynomial of degree $> 3$ is irreducible?
What's the easiest way to check if a polynomial of degree > 3 is irreducible in $\mathbb{Z}_2[x]$?
I want to find out if $x^7+x^6+1$ is irreducible in $\mathbb{Z}_2[x]$.
If a quadratic polynomial ...
