Questions tagged [field-theory]
Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.
10,081
questions
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1answer
10 views
Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?
We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. ...
0
votes
0answers
23 views
Integral closure of a field with uneven characteristic
I am studying a lof of algebra lately and i am stuck on this problem.
Let $K$ be a field with uneven characteristic and $E = K(X)(\alpha) $ where $\alpha^2= X^2 -1 \in K(X)$
I am struggling to find ...
0
votes
0answers
31 views
Find splitting field of $(x^3-x^2-x)(x^4-x^2+1)$ over $\mathbb{F}_3$
As written in the title, I have to compute the splitting field of $$(x^3-x^2-x)(x^4-x^2+1)$$ over $\mathbb{F}_3$
I'd like to understand if my attempt is correst, or if I'm missing something. Here's ...
0
votes
2answers
36 views
Integer basis for $\mathbb{Z}_{\mathbb Q(\alpha)}$
So we are studying integral field and i am stuck on this problem.
Let $f(X) = X^3 -X -12 \in \mathbb{Q}[X]$ and $\alpha \in \bar{\mathbb{Q}}$ be a zero of $f(X)$.
I already showed that $f(X)$ is ...
0
votes
1answer
30 views
Find irreducible factors without factorizing [closed]
I have an exercise from my course notes that states:
Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it)
I see immediately that the $...
0
votes
1answer
11 views
Confusion Over Polynomial Separability Example
In lecture notes I am reading for Abstract Algebra, in the Separability section for Field Theory, it gives the introduction and definition that a separable polynomial is one that has no repeated roots ...
1
vote
0answers
22 views
Convergent power series and Euclidean domain, or field.
Consider the following power series (as a function of $z$):
$$G(z) = \sum_{i=0}^\infty a_i z^{-i},$$
where $a_i$ is a real coefficient and $z$ is any complex number. It also satisfies that
$$\sum_{...
0
votes
1answer
23 views
Number of maximal ideal in the quotient ring [closed]
How to find the number of maximal ideal in the Quotient ring $\frac {\mathbb{Z}_5{[x]}}{<(x+1)^2(x+2)^3>}$. ?
$R/A$ is an integral domain iff $A$ is prime.
1
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1answer
24 views
Is there an isomorphism of fields between $\mathbb{F}_{3^{2}}$ and $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$?
if $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$ where $i=\sqrt{2}=\sqrt{-1}$ and we define $(a+bi)+(c+di):=(a+c)+(b+d)i$ and $(a+bi)\ast (c+di):=(ac-bd)+(ad+bc)i$
Is there an isomorphism of fields ...
0
votes
1answer
27 views
Unique subfield $L$ of $K=\mathbb{Q}(\sqrt[5]{2},\zeta_5)$ such that $[K:L]=5$
$K$ is the splitting field of the polynomial $f=x^5-2$, and we need to prove the existence of a unique subfield $L$ such that $[K:L]=5$. It would be nice to use the Galois group here, but it is a semi-...
2
votes
1answer
13 views
Definition of non split Cartan subgroup
What is the definition of the non split Cartan subgroups of $GL_2(\mathbb{F}_p)$?
And what are the explicit expression of a matrix of this subgroups?
I read on "Modular Functions of One Variable III" ...
-1
votes
3answers
57 views
Proving F is a field [closed]
Let $F=\{a+bi; a,b \in F_3\}$ where $i=\sqrt{2}=\sqrt{-1}$
and we define $(a+bi) + (c+di) := (a+c)+(b+d)i$ and $(a+bi) * (c+di):= (ac-bd)+(ad+bc)i$ with $0=0+i$
and $1=1+0i$
Prove $F$ is a field
1
vote
0answers
11 views
Show this linear map over a finite field is not the zero map [duplicate]
Let $q=p^k$ be a prime power, let $m\in\mathbb{N}$ and consider the map
$$
\tau:\mathbb{F}_{q^m}\longrightarrow\mathbb{F}_{q^m},\quad a\longmapsto\sum_{i=0}^{m-1} a^{q^i}.
$$
Show that $\tau$ is ...
1
vote
4answers
40 views
For which $n$ is $\sum_{i=0}^n x^i\in\mathbb{Q}[x]$ irreducible?
I know that when $n+1=p$ is prime, $f=\sum_{i=0}^{p-1} x^i$ is the minimal polynomial of $\zeta_p$, a primitive $p$-th root of unity, hence is irreducible. This can be shown by applying Eisenstein to $...
9
votes
4answers
357 views
Understanding Fraleigh's proof of: Every finite integral domain is a field
Here's how Fraleigh proves: Every finite integral domain is a field in his book:
Let
\begin{equation*}
0, 1, a_1, \dots, a_n
\end{equation*}
be all the elements of the ...
1
vote
1answer
22 views
Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$
Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ ...
1
vote
1answer
32 views
Degree of extension $\mathbb{C}/K$, where $K$ is maximal with the property $\sqrt{2} \notin K$
This question has been asked before but not really answered, but my query is a bit separate. To summarise the details: $K$ is a field maximal with respect to the property $\sqrt{2}\notin K$, any ...
2
votes
2answers
104 views
If the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?
I have a question: if the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?
0
votes
0answers
28 views
Degree of an exclusionary Field Extension
Let's say I've got a field $\mathbb{Q}[i]$\ $\mathbb{Q}$. What's the degree of the field extension $\mathbb{Q}[i]$\ $\mathbb{Q}$ : $\mathbb{Q}$? Clearly without the exclusion this has a degree of 2; ...
1
vote
1answer
32 views
$f,g \in k[t]$ with $k(f,g)=k(t)$, $\deg(f)=2$ and $\deg(g)=3$
Let $f=f(t),g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
Assume that the following two conditions are satisfied:
(i) $\deg(f)=2$ and $\deg(g)=3$.
(ii) $k(f,g)=k(t)$.
Question: Is it ...
2
votes
0answers
31 views
Why a polynomial is irreducible over Z
Prove that the following polynomial is irreducible over $\mathbb{Z}$:
$$f(x) = x^8-x^7+x^5-x^4+x^3-x+1$$
My attempt: one can see that $f(x)=(x^4+x^3+1)(x^4+x+1) $ over $\mathbb{F_2}$ where these 2 ...
1
vote
1answer
32 views
Galois group of field extension
I was asked to find the Galois group of the extension $\mathbb{Q}(\sqrt[3]{2},\sqrt{2},e^{\frac{2\pi i}{3}})$. Since the degree of the minimal polynomials of $\sqrt[3]{2},\sqrt{2}$ and $e^{\frac{2\pi ...
0
votes
1answer
49 views
Finding a minimal polynomial of a root of unity over a field extension
I'm trying to find the minimal polynomial of the seventh root of unity over the field $Q(i\sqrt{7})$. I know how to do this over the rationals and have proceeded to finding that $(x-1)(x^6 +x^5 +x^4 +...
1
vote
1answer
73 views
$f,g \in k[t]$ such that $\deg(f)=\deg(g)$ and $k(f,gt)=k(t)$
Let $f=f(t), g=g(t) \in k[t]$ be two nonzero polynomials over a field $k$ of characteristic zero.
Assume that the following two conditions are satisfied:
(i) $\deg(f)=\deg(g) \geq 1$.
(ii) $k(f,gt)=...
2
votes
1answer
28 views
How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$
This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$.
After reading this, I am still confused on how to find ...
-1
votes
1answer
33 views
Write de addition and multiplication tables for this field [closed]
F3 is the set of residue class of 3
how can i do the multiply and addition table for this field of 9 elements
0
votes
0answers
26 views
Does adjoining an element to a field change the characteristic? (No it doesn't.)
This may have an answer somewhere already, but I can't find it. Let $K$ be a field of characteristic $p$. Let's adjoin, for example, $\zeta_n$ where $\zeta_n$ is some primitive $n$-th root of unity. ...
-1
votes
2answers
34 views
(Proof verification) there is no homomorphism between a finite fieldās additive group to its multiplicative group.
Given a finite field F with additive group $\text{F}^+$ and multiplicative group $\text{F}^{\times}$ Show that there doesnāt exist $f$:$\text{F}^+ \to \text{F}^{\times}$ s.t. $f(x+y)=f(x)f(y)$.
Proof ...
0
votes
1answer
41 views
If $K(\zeta,\beta)/K$ is the splitting field for $f$, what can we say about $K(\zeta,\beta)/K(\zeta)$?
Problem setup:
Let $n\in\mathbb{N}$ and let $K$ be a field whose characteristic does not divide $n$. The splitting field of $f=x^n-c$ ($c\not=0$) over $K$ is $K(\zeta,\beta)$ where $\zeta$ is a ...
0
votes
1answer
25 views
Equivalence between integral closures
I am struggling with the following problem. Let $S$ be a Integral domain and $K$ its Quotient field. Furthermore let $A/B/K$ be Field extensions.
We set $X = \text{Int}_S(B)$ where $\text{Int}_S(B)$ ...
0
votes
0answers
13 views
Question regarding p-basis in complete local ring
Let $k$ be the residue field of characteristic $p>0$ of a complete local ring $R$ with maximal ideal $m$. A $p$-basis for $k$ is a subset $B \subset k$ such that
(i) $k^p(B)=k$
(ii) all ...
0
votes
1answer
33 views
Finding class number of quadratic number field using Minkowski bound
My understanding of this is as follows:
In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$.
Minkowski ...
0
votes
1answer
29 views
Real or imaginary part of a field extension of $\mathbb{Q}$.
When $L/\mathbb{Q}$ is a finite extension of $\mathbb{Q}$ that doesn't lie in $\mathbb{R}$. Is there anything we can say about the degree of the extension $L/ L \cap \mathbb{R}$?
I know that for ...
0
votes
1answer
30 views
$a$ is a norm in $K(\sqrt{a})/K$ if and only if it is the sum of two squares
$a$ is a norm in a quadratic extension $K(\sqrt{a})/K$ if and only if $a=u^2+v^2$ for some $u,v\in K$. $K$ an arbitrary field.
Only if follows from:
If $a$ is a sum of squares then there exists an ...
1
vote
0answers
37 views
Find basis of $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$ over the field $\mathbb Q(\sqrt[3]{2}\omega)$.
Splitting field for the polynomial $\ x^3-2$ over $\mathbb Q$ is $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$. Now roots of the above polynomial are $\sqrt[3]{2}, \sqrt[3]{2}\omega, \sqrt[3]{2}{\omega}^2$
Since ...
1
vote
1answer
31 views
Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.
Let $K$ be a field of characteristic $p > 0$, and $f ā K[X]$ monic and irreducible with root $\alpha$. Let $F$ be the Frobenius endomorphism.
To demonstrate: $\alpha ā F[K(\alpha)] \Rightarrow f ā...
1
vote
3answers
67 views
Galois group of $x^p-a$ over $\mathbb{Q}$
I have found that the Galois group $G$ of $f=x^p-a$ over $\mathbb{Q}$ is of order $p(p-1)$. I need to show that if $P$ is a subgroup of $G$ of order $p$, then $P$ is normal and $G/P$ is cyclic. ...
8
votes
2answers
955 views
Which field property enables us to multiply on both sides by the same value, while preserving equality? [duplicate]
I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off.
I was ...
0
votes
1answer
41 views
Calculate order of multiplicative group of finite field
How can one calculate the order of a multiplicative group of a finite field such as:
$(\mathbb{F}(2^3) \backslash \{0\}, \times)$
Is it as simple as doing $2^3-1$ ?
1
vote
1answer
24 views
Is every field a field extension of some form.
I am new to finite field theory .While I was going through the theory I figured that $\mathbb{C}$ is in fact $\mathbb{R}(i) $ isomorphic to $\mathbb{R}[x]/(x^2+1)$ .So I had a question in mind is ...
0
votes
0answers
42 views
Can Field of fractions be Algebraically Closed?
Let R be an integral domain which isn't a field.
Can it be the case that the field of fractions is algebraically closed?
The reason I'm asking this is : The field of fractions of $\mathbb Z$ is $\...
3
votes
3answers
76 views
Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$
Let $\mathbb{F}_q$ denote a field with $q=p^n$ elements, where $p$ is prime. Consider the polynomial $f=x^{p^{n-1}}+\ldots+x^p+x$ and the sets
$$
\begin{align*}
S&=\{a^p-a:a\in\mathbb{F}_q\},\\
...
3
votes
0answers
99 views
Galois Group of $x^{6}-2x^{3}-1$
I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt[3]{1+\sqrt{2}}$.
I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
0
votes
1answer
20 views
Express the fixed field of a subgroup $H$ of $\Gamma(L:K)$ in terms of a basis of $L$ over $K$.
I've been trying to solve this problem for $L:K$ finite with char $K = 0$:
$\beta_1,...,\beta_n$ basis for $L$ over $K$, $H$ subgroup of $\Gamma(L:K)$ $\implies$ $\phi(H) = K(\gamma_1,...,\gamma_n)$, ...
0
votes
1answer
28 views
Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ā[x]$ if and only if $f({d\over2})ā 0$ for all $a, b, c, dāā¤$.
$f(x)=2x^3+ax^2+bx+c$ where $a, b, cāā¤$. Prove that f is irreducible in $ā[x]$ if and only if $f({d\over2})ā 0$ for all $dāā¤$.
Using a hint from my professor I have attempted a proof but I don't think ...
1
vote
1answer
38 views
$X^{p^k} - a ā K[X]$ irreducible?
Let $K$ be a field of characteristic char$(K) = p > 0$, and let $a ā K$ be an element with the following property:
$$(\forall \beta ā K)(\beta^p ā a).
$$
Let $k ā ā$ be arbitrarily given. Is it ...
-2
votes
1answer
39 views
How to show $E\otimes_k\bar k$ has at least two prime ideals? [closed]
Suppose $k\subsetneqq E$ are two fields, and $E$ is separable over $k$, $\bar k$ is the algebraic closure of $k$, how to show $E\otimes_k\bar k$ has at least two prime ideals?
1
vote
1answer
48 views
Product of the elements in Galois group and irreducible polynomial
Statement) Let $K$ be a Galois extension over a field $F \ \text{s.t.} \ \vert G(K/F) \vert =[K;F] =n $
Say $ G(K/F) = \{ \sigma_1(=id) , \sigma_2,..., \sigma_n \}$
Take a element of $\alpha \in K ...
-1
votes
1answer
36 views
Use of irreducible polynomial in finite field construction
When constructing a finite field $\mathrm{GF}(p^n)$ using polynomials:
Why do we need to modulo an irreducible polynomial? What happens if this polynomial is reducible?
Does such an irreducible ...
3
votes
1answer
62 views
Find an irreducible polynomial in $ Q[x]$ of degree $726$.
Find an irreducible polynomial in $ Q[x]$ of degree $726$.
I first thought of $x^{726}+1$ to start but its roots would be complex so not in $Q$.
Now I'm thinking to use Eisenstein, so taking the ...
