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.
11,047
questions
0
votes
0answers
10 views
Proof that the following polynomial is irreducible
I need to check whether the following polynomial $g(x)=x^2+t^2-1\in\Bbb{C}(t)$ (where $\Bbb{C}(t)$ is the field of all rational expressions over $\Bbb{C}$) is reducible. My attempt: I assume by ...
0
votes
0answers
10 views
Intermediate extension of degree n of the algebraic closure of Q [duplicate]
I need to prove that for every natural number n, there exists an intermediate extension E of A/Q with [E:Q]=n, being A the algebraic closure of Q (the field of all complex numbers that are algebraic ...
0
votes
1answer
26 views
Existence of a non-zero T- invariant subspace on $\mathbb{Q}^{4}$
For any linear transformation $T:\mathbb{Q}^{4} \to \mathbb{Q}^{4} $, does there always exist a non-zero T- invariant subspace?
As we know
For any linear transformation $T:\mathbb{R}^{4} \to \mathbb{...
0
votes
1answer
19 views
Zeroes of $x^{p^n} - x$ are distinct in splitting field of $\mathbb{F}_p$
I am attempting to show that the zeroes of $x^{p^n} - x$ are distinct (as a part of showing the splitting field over $\mathbb{F}_p$ has $p^n$ elements). I understand that it is sufficient to show that ...
2
votes
2answers
56 views
When can we embed field extensions into one another?
I was recently informed of the following theorem:
Let $K$ be a field of characterisic $0$, and let $L$, $M$ be field extensions of $K$ such that
i) $L$ is algebraic over $K$,
ii) every irreducible ...
0
votes
1answer
36 views
Prove $\text{Hom}(K,K)=\{0,id_K\}$ for $K=\mathbb{Q}$
I've been solving some problems from my Galois Theory course, and I need help with the final step of this one:
Prove $\text{Hom}(K,K)=\{0,id_k\}$ for $K=\mathbb{Q}$.
The work I did so far:
Given $K$ ...
4
votes
2answers
45 views
For which rational coefficients is $a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4$ constructible, where $\alpha=3^{1/5}$?
Let be $\alpha=3^{1/5}$, and be $\gamma$ the number
$$
\gamma=a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4
$$
with $a,b,c,d,e \in \mathbb{Q}$ . For which $a,b,c,d,e \in \mathbb{Q}$ is $\gamma$ ...
0
votes
0answers
18 views
Does maximality of $p=A \cap q$ imply maximality of $q$, where $A \subseteq B$ is algebraic?
Let $A \subseteq B$ be two commutative $\mathbb{C}$-algebras.
Let $q$ be an arbitrary ideal of $B$ and denote $p:=q \cap A$.
By [Lemma 3][1], if $A \subseteq B$ is an integral extension, then: $q$ is ...
2
votes
1answer
51 views
Showing that $\mathbb{Q}(\sqrt{3+\sqrt{3}})$ is not Galois.
Why $\mathbb{Q}(\sqrt{3+\sqrt{3}})$ is not Galois?
I have only this: Let $\alpha=\sqrt{3+\sqrt{3}}.$
If $\alpha=\sqrt{3+\sqrt{3}}$ then $(\alpha^2-3)^2=3$ then $f(x)=x^4-6x^2+6$ is the minimal ...
0
votes
0answers
33 views
An algebraic structure that enables solving radical equations rigorously using that algebraic structure operations only.
My understanding of abstract algebra, and the concept of a field in particular, is that it's an abstract algebraic construction that resembles the real numbers. And that it abstracts out and captures ...
0
votes
0answers
40 views
Question on Group Actions and one to one correspondence with a field
I am really struggling with this problem: Let $\mathbb{F}$ be a field and $\mathbb{F}^{\times}$ is a group under multiplication that consists of the units in $\mathbb{F}$. We consider the following ...
1
vote
2answers
35 views
A cubic polynomial over $\mathbb{Q}$ with a degree three splitting field?
I'm wondering if it's possible to have a cubic polynomial $f(x)\in\mathbb{Q}$ with three distinct roots (in $\mathbb{C}$) that has a degree three splitting field?
If $f(x)=(x-a)(x-b)(x-c)$, with $a,b,...
2
votes
1answer
27 views
Given examples of complete non-Archimedean field of characteristic $p$ s.t. $K$ is not $F$-finite
Let $K$ be a complete non-Archimedean field of characteristic $p$ with valuation ring $A$. We say a ring $R$ is $F$-finite if it is of characteristic $p$ and the absolute Frobenius map $F_R$ is finite....
-2
votes
0answers
49 views
I have read that any integral domain can be extended to be a field. How do I do it?
There is an associative, commutative infinite-dimensional non-Archimedean algebra over $\mathbb{R}$ with no zero-divisors (integral domain), where we cannot divide by some elements.
If we try, we get ...
1
vote
1answer
70 views
Adjoing $p$th roots to rings: as quotients of polynomials
Motivation/Setup: Suppose $R$ is a domain. I'd want to "adjoin $p$th roots of an element $a \in R$." where $p$ is a prime (which i am mostly interested in $p$-adic) This seems to be a cmomon ...
1
vote
2answers
40 views
Group of units in a finite field
Let $f(x) \in \mathbb{Z}/3\mathbb{Z}[x]$ be a cubic irreducible polynomial and let $F = \mathbb{Z}/3\mathbb{Z}[x]/(f(x))$. I want to show that either $x$ or $2x$ generates the group of units of $F$.
...
0
votes
0answers
37 views
How to define operations $+$and $\cdot$ if a set has $3$ elements?
How to define operations $+ $and $\cdot$ if a set $F$ has $3$ elements {j, k,l } rather than the usual $2$ elements using the table so that it is a field? And more broadly, how can I define operation ...
0
votes
0answers
30 views
Why does the Nullstellensatz imply that there is a zero of a maximal ideal which is algebraic over a field?
I am reading Ernst Kunz's "Introduction to commutative algebra and algebraic geometry" and I'm not quite understanding why the underlined sentence in the screenshot below is true.
The ...
4
votes
2answers
88 views
Prove that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{3})$
Showing that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})$ has degree 6 over $\mathbb{Q}$ is straighforward:
It contains $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{3})$ which are degree 2 and 3 over $\...
1
vote
1answer
37 views
What is $[\mathbb{Q}(a) : \mathbb{Q}(a+1/a) ]$?
I've been thinking about this for a while and I keep confusing myself, so hopefully someone can help clarify some thoughts.
I'm currently looking at two fields, $\mathbb{Q}(a+1/a)$ and $\mathbb{Q}(a)$,...
3
votes
0answers
24 views
Basic question about inseparable field extension
Suppose $L/K$ is not separable. Does this imply that there is an $a\in L$ such that $a^p\in K$ but $a\not\in K$?
I know that the characteristic of the fields must necessarily be $p>0$. By ...
1
vote
1answer
51 views
Möbius transformation preserving polynomial
Let $K$ be an algebraically closed field, and $x$ transcendent over $K$. We know that the group of $K$-automorphism of $K(x)$ is $\operatorname{PGL}(2,K)$, whose elements satisfy $ \alpha (x)= \...
1
vote
1answer
40 views
Countable algebraically closed field inside an incountable algebraically closed of characteristic zero
I would like to know if the following claim is true. I found this claim in a paper without proof :(.
Let $k$ an uncountable algebraically closed field of characteristic $0$.
Let $S=\operatorname{Spec}...
1
vote
1answer
19 views
K-isomorphic splitting fields in field with characteristic 2
The problem is as follows:
Let $K$ be a field of characteristic $2$, let $a$ be an element of $K$ which is not of the form $b^2 + b$ for any $b \in K$, let $f_a(X) = X^2 + X + a$, and let $L_a = K(\...
1
vote
0answers
19 views
Finding minimal polynomial of an element in an extension over $\mathbb{Q}$
What is the minimal polynomial of $\sqrt2 +\sqrt{2}i$ over the field $\mathbb{Q}$.
I proceeded in the following way:
Let $x=\sqrt2+\sqrt{2}i \Rightarrow x^2=2+2i^2+4i \Rightarrow x^2=4i \Rightarrow x^...
1
vote
0answers
19 views
intersections of powers of maximal ideal in integral domain
Let $\Omega\subset\mathbb{C}$ be a disc. $p\in \Omega$.
In the integral domain $R=H(\Omega)$ " the space of holomorphic functions in $\Omega$, consider the maximal ideal $M_p=\{f \in R : f(p)=0\}$...
1
vote
1answer
15 views
When does conjugation in simple field extensions coincide with the identity?
Suppose I have a field $F$ and a simple extension of $F$, $F(\alpha)$, and let's assume that $\alpha^2 \in F$. Now take the "conjugation" automorphism $\sigma:F(\alpha) \to F(\alpha)$, $\...
0
votes
0answers
26 views
Degree of $\mathbb{Q}(\sqrt{a_1}, \ldots , \sqrt{a_n})$ over $\mathbb{Q}$ for $a_i$ squarefree and $gdc(a_i,a_j)=1$
It seems obvious that $[\mathbb{Q}(\sqrt{a_1}, \ldots , \sqrt{a_n}):\mathbb{Q}]=2^n$,provinding that $a_i$ is squarefree $i=1, \ldots, n$ and $gcd(a_i,a_j)=1, \forall i \neq j $but I am having ...
0
votes
3answers
22 views
Given f(x) in k[x], is it possible to find two field extensions K/k and K'/k such that f has two different factorizations as linear polynomials?
By Kronecker's theorem, given a field $k$ and $f(x)$ a polynomial with coefficients in $k$, there exists a field $K$ containing $k$ as a subfield and with $f(x)$ a product of linear polynomials in $K[...
6
votes
1answer
76 views
Splitting field of a polynomial over Z_5
Find the splitting field of the polynomial $$x^3+x+1$$ over $\mathbb{Z}_5$.
I can see that the given polynomial is irreducible over $\mathbb{Z}_5$ So, $$\dfrac{\mathbb{Z}_5 [x]}{(x^3+x+1)}$$ is a ...
-3
votes
1answer
60 views
What primes factor both $2$ and $3$, with no restriction on the domain? [closed]
What numbers (especially primes) factor both $2$ and $3$ (with no restriction on the domain)? How many answers are there? I'm looking preferably for a prime.
My attempt:
No natural number (other ...
-1
votes
0answers
44 views
How would I determine all the cubes in mod 13? [closed]
Just looking how to do this I have been asked to find all cubes in Z_13, which I assume is cubing all elements of Z_13 under mod 13, then what is the remainder is what can be involved in the set?
0
votes
0answers
40 views
Polynomial of degree $3$ from $F[x]$ has a zero in $F$, then how can I know about its reducibility? [closed]
I have a polynomial of degree greater than $3$ from $F[x]$, F is field has a zero in $F$, then how can I know about its reducibility and irreducibility?
e. g, $x^5+x-3$ irreducible or not over $R$?
I ...
2
votes
1answer
36 views
Suppose $F ā K$ are fields. Let $f(x) ā F[x] ā K[x]$. Suppose that $f(x)$ is irreducible in $K[x]$. Prove that $f(x)$ is also irreducible in $F[x]$.
Suppose $F ā K$ are both fields. Let $f(x) ā F[x] ā K[x]$. Suppose that $f(x)$ is irreducible in $K[x]$.
$a)$ Prove that $f(x)$ is also irreducible in $F[x]$.
$b)$ Is it true that if $f(x)$ is ...
0
votes
1answer
28 views
Normal open subgroup of $Gal (L / K)$
Let $L / K$ be Galois extension and $H$ be a subgroup of $Gal (L / K)$.
Suppose that $K$ is a subfield of $L$ that is invariant by $H$.
Show that $HN = Gal (L / K)$ for any normal open subgroup $N$ of ...
2
votes
1answer
50 views
Can the minimal polynomial have multiple roots?
Let $F$ be an extension field of $K$. I'm trying to think of an example of an element $\alpha$ which is algebraic over $K$ and whose minimal polynomial over $K$ has a root of multiplicity $> 1$ in ...
1
vote
0answers
20 views
Understanding why $E(\alpha)$ is a splitting field for $F(\alpha)$ when $E$ is for $F$.
Why if $E$ is a splitting field of a polynomial $f$ over $F$ and $\alpha$ in some extension of $F$ then $E(\alpha)$ is a splitting field of $f$ over $F(\alpha)$?
The argument that justifies the above ...
-3
votes
0answers
41 views
Is every zero divisor free ring a field? If that's not the case give an example. [closed]
Is every ring without zero divisors a field? If that's not the case give an example.
A zero divisor exist if $x\cdot y =0$, while $x\neq 0 $ and $y\neq 0$.
As I'm studying for an upcoming exam I have ...
0
votes
0answers
32 views
Polynomial reduction modulo p
I'm trying to read this research paper, but there's a section I can't understand. So we have this polynomial:
$Q_{k}(z)=z^{6k+7}ā8z^{6k+6}+ 26z^{6k+5}ā44z^{6k+4}+ 41z^{6k+3}ā20z^{6k+2}+ 4z^{6k+1}+ 15z^...
0
votes
0answers
22 views
Vector spaces and fields, addition and product operators.
Thank you for taking the time to read my question. I am reading about fields and vector spaces. Any vector space $V$ is a vector space over some field $F$. Furthermore, for any set $S$, it is my ...
2
votes
3answers
55 views
Field extension of $\Bbb Q$ of degree $4$ has reals
Suppose $L / \mathbb{Q}$ is a field extension with $[L : \mathbb{Q}]$ = 4, with $L \not\subset \mathbb{R}$. Is it true that $L \cap \mathbb{R} \neq \mathbb{Q}$? If not, is it true if $L / \mathbb{Q}$ ...
1
vote
2answers
48 views
Let $F$ be a field. Prove that $F$ cannot have two different additive identities [closed]
Let $F$ be a field. Prove that $F$ cannot have two different additive identities (i.e., there cannot be two different elements $a, b \in F$ such that, for every $x \in F$, $a+x = x+a = x = b+x = x = b$...
0
votes
1answer
77 views
Why is $F=\{0\}$ a field?
Context: I am new to this. I started my course yesterday.
I know that the operations $+$ and $\cdot$ are required to satisfy the field axioms. So how can $F=\{0\}$ be a field?
Recall the additive and ...
1
vote
0answers
41 views
Lie groups over fields of finite characteristic
Does anyone have any good references on Lie groups over fields of finite characteristic? I am trying to find something comprehensive that shows what fails and what succeeds in comparison to Lie theory ...
0
votes
1answer
19 views
Fixed field with non-normal subgroup and Galois property
Let $L/K$ be a finite Galois extension with Galois group $G$. Furthermore, let $g \in G$ and $M = L^{\langle g \rangle}$.
I know that $\langle g \rangle$ does not always have to be a normal subgroup ...
0
votes
1answer
63 views
What are the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$?
I would like to calculate the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$.
I know that the elements of $\mathbb{Q}(\sqrt[3]{2})$ have the form of ${a+b\sqrt[3]{2}+c\sqrt[3]{4}}$, where a,b,c $\in \...
2
votes
0answers
42 views
What is the intuition behind the definiton of a seperable field extension?
so I am currently studying separable extensions in John Fraleigh's abstract algebra book and have come upon this definiton of a separable field extension:
A finite extension $E$ of $F$ is a separable ...
3
votes
1answer
52 views
A property of fields of characteristic $p$ and their extensions
I am currently stuck on the following problem, taken from Herstein's Topics in Algebra, 2nd ed.. It reads:
If $F$ is of characteristic $p \neq 0$ and if $K$ is a finite extension of $F$, prove that ...
0
votes
1answer
31 views
Relation between $\mathbb{C}$-automorphisms and Möbius transformations.
Today in my Algebraic Ecuations course, the proffesor mentioned that while there is only one field automorphism (the trivial one) in $\mathbb{R}$, there are infinite field automorphisms in $\mathbb{C}$...
1
vote
0answers
31 views
Which elements of $\mathbb{R}[x]$ have multiplicative inverses ? And is $\mathbb{C}[x]$ a field?
Let $\mathbb{R}[x] = \{a_n x^n + \cdots + a_0 \mid \text{where each $a_i \in \mathbb{R}$ and $n \in \mathbb{N} \cup \{0\}$}\}$.
Let $\mathbb{C}[x] = \{a_n x^n + \cdots + a_0 \mid \text{where each $a_i ...
