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.
0
votes
1answer
22 views
Complex conjugate and Field Extension
Let $E$ be a subfield of $\mathbb{C}$ and Let $\overline{E}=\{\overline{z} \, |\, z \in E \}$ with $\overline{z}$ being the complex conjugate of $z$. Let $K$ be a subfield of $\mathbb{C}$ with $\...
0
votes
1answer
9 views
How to show that this simple extension avoids adding more elements in $L$?
$F\subset L \subset K$ is a field extension, $α \in K$ is algebraic element over $F$ , whose minimal polynomial $p(x)$ over $F$ is irreducible over $L$, show that $F(α)\cap L = F$
I don't know how to ...
1
vote
0answers
40 views
Does there exist an automorphism of $\mathbb{C}$ that does not keep $\mathbb{R}$ fixed? [duplicate]
I'm going to take Galois theory in the upcoming semester and I'm working on some basic problems right now. I was trying to figure out what $\mathrm{Aut}(\mathbb{C})$ should be. If $f: \mathbb{C} \to \...
2
votes
3answers
58 views
Proof verification: Proving or disproving that $(\Bbb R^+, \oplus, \otimes)$ as defined is a field
The problem I have been working on, in summary:
Let $R$ denote the set of positive real numbers. Define addition, denoted $\oplus$, and multiplication, dentoed $\otimes$, respectively by $a \oplus ...
2
votes
1answer
44 views
Is this a counterexample?
Suppose $K $ is a field and $\overline K $ an algebraic closure. Let $f $ be a $K $-automorphism of $\overline K$, let $L$ be the subfield of $\overline K $ fixed by $f $. In this post : (link), they ...
0
votes
0answers
19 views
Algebraic Field Extensions and Irreducible Polynomials
Let $E/K$ be a field extension, $a,b\in E$ algebraic over $K$. Show:
$\text{Min}(a,K,X)$ irreducible over $K(b)$ if and only if $\text{Min}(b,K,X)$ irreducible over $K(a)$
My attemp:
(1) Let deg $\...
0
votes
1answer
47 views
find all algebraic elements of $\mathbb{Q}(\pi)$ over $\mathbb{Q}(\pi^2 -2\pi +5)$
Well, it is pretty clear that $\mathbb{Q}(\pi^2-2\pi+5)$ are all algebraic. I don't have any idea how to justify it, but I think those are the only algebraic elements of $\mathbb{Q}(\pi)$ over $\...
2
votes
1answer
26 views
Field extensions and Intermediate Rings
Let $E/K$ be a field extension. Show the following equivalence:
(i) E/K is algebraic
(ii) Every Intermediate Ring of $E/K$ is a Field.
Here is what I tried:
(i)$\Rightarrow$(ii) Let $E/K$ be ...
4
votes
4answers
58 views
Showing existence of irreducible polynomial of degree 3 in $\mathbb{F}_p$
I'am trying to show that for every p$ \in \mathbb{N}$ where p is prime,
there is an irreducible polynomial of degree 3 in $\mathbb{F}_p$.
I've found too general answers for that question, but I want ...
-2
votes
0answers
27 views
Algebraic Closure - equivalent definitions [on hold]
I want to show that the following characterizations of the algebraic closure are equivalent. Let K be a field
(i) $E=K$ for every algebraic extension $E/K$
(ii) Every non-constant Polynomial in $K[...
0
votes
2answers
68 views
What is the minimal polynomial of $\sqrt{3+4i}+\sqrt{3-4i}$ over $\mathbb{Q}$?
My attempt:
I know that $P=\mathbb{Q(\sqrt{3+4i}+\sqrt{3-4i})} \subseteq \mathbb{Q(\sqrt{3+4i},\sqrt{3-4i})}=K$. But I don't know whether $K \subseteq P$. Assuming $K=P$. I can see that K is Galois ...
0
votes
0answers
49 views
How to find a basis of $\Bbb R(t)$ over $\Bbb R$?
I am reading field theory.
I dont understand the following fact:
What will be a basis of $\Bbb R(t)=\{\frac{f(t)}{g(t)}:f(t),g(t)\in \Bbb R[t]\}$.
I know that $\Bbb R(t)$ is a field and hence a ...
2
votes
1answer
21 views
Example of an infinite simple extension.
I'm new to Field Theory and I'm looking for an example of an infinite simple extension.
Theorem: The element $\alpha$ is algebraic over $F$ if and only if the simple extension $F(\alpha)/F$ is ...
2
votes
2answers
72 views
Why must an automorphism of an extension of $\mathbb{Q}$ send 1 to a rational number?
This is from a video I was watching that claimed this:
If $\phi$ is an automorphism of an extension field $F$ of $\mathbb{Q}$, then $\phi(q)=q$ for all $q\in\mathbb{Q}$.
The proof started by ...
5
votes
2answers
95 views
Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$
I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
0
votes
0answers
25 views
Compute the Galois group of L over K [duplicate]
$\textbf{Problem}$ $K=\mathbb{Z}/p\mathbb{Z}$; L=splitting field of $\prod_{i=1}^{p-1} (t^2-i)$, where $p$ is an odd prime
$\textbf{Attempt}$ Since $(-k)^2=(p-k)^2=k^2$, we know that there are $\...
0
votes
1answer
19 views
Degree of splitting fields
I'm learning about splitting fields but I'm not sure if I am right. Hopefully I can get some insights on whether I have been learning correctly.
The question asks to find the degree of the splitting ...
1
vote
2answers
48 views
Describe $\Bbb{R}[x]/(x^2 + 1)$
First by previous knowledge, I do know that $\Bbb{R}[x]/(x^2 + 1) \cong \Bbb{C}$, so this might seem trivial. But I am not here to ask about that and I don't want to use this fact
I am reading Dummit ...
0
votes
0answers
3 views
Question regarding separable extension
If $K$ is a field extension of $F$ and if $\alpha$ $\in$ K is not separable over F, show that $\alpha^{p^{m}}$ is separable over F for some $m\geq 0$. Where $p=char(F)$.
I have been trying it for ...
0
votes
0answers
34 views
Number Fields with Cyclic Galois Groups
Let $\xi\in\overline{\mathbb{Q}}\backslash\mathbb{Q}$
be an algebraic number of degree $d\geq2$
and let $\mathbb{K}=\mathbb{Q}\left(\xi\right)$.
What are some conditions on $\xi$
and $d$
...
4
votes
2answers
104 views
Turning $\mathbb R^n$ into field
I am reading Apostol's fascinating text Mathematical Analysis. In a footnote on P117, he writes:
If it were possible to define multiplication in $\mathbb R^3$ so as to make $\mathbb R^3$ a field ...
3
votes
1answer
31 views
Let K be the splitting field of the polynomial $x^4-2x^2-2$. Find an automorphism $\sigma \in Gal(K/\mathbb{Q})$ of order 4.
My attempt:
The roots of the polynomials are $x=\sqrt{1+\sqrt{3}}, -\sqrt{1+\sqrt{3}}, -\sqrt{1-\sqrt{3}}, -\sqrt{1-\sqrt{3}}$. and the automorphisms $\sigma \in Gal(K/\mathbb{Q})$ permutes the root ...
1
vote
1answer
58 views
Major misunderstanding about field extensions and transcendence degree
So presumably this question is very basic, but I'm having some trouble with apparent contradictions in my reasoning.
Let $k$ be a field and $k \subseteq K$ a field extension. We say that $K$ is a ...
0
votes
0answers
27 views
Given a valuation on a field , does it always extend to a valuation on every extension field
Let $(G,+,\ge)$ be a totally ordered abelian group. Let $K\subseteq L$ be an extension of fields. Let $v : K\setminus \{0\} \to G$ be a valuation (https://en.wikipedia.org/wiki/Valuation_(algebra)) .
...
0
votes
0answers
34 views
Agebraically closed field equivalent definition
Is this a valid equivalent definition of an algebraically closed field?
Let $ F $ be a field. Define a function $ f(x): F\longrightarrow F $ as a finite combination of the operations in $ F $ (...
1
vote
0answers
41 views
Affine Varieties over separably closed fields
Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$.
On the ...
-3
votes
1answer
17 views
Algebraically independent numbers and Archimedean field
What is the cardinality of the set of all algebraically independent numbers in $\mathbb{R}$?
Can this be related to the total number of Archimedean fields possible as rational extensions of sets of ...
4
votes
2answers
57 views
What does it mean that field $\mathbb{F}_{p^n}$ “contains” the prime field $\mathbb{Z}_p$?
I have read in few books (example Computational Number Theory, page 77) that any extension field $\mathbb{F}_{p^n}$ "contains" as a subfield the prime field $\mathbb{Z}_p$?
What exactly does "...
2
votes
2answers
35 views
Prove that if $p \not\equiv 1 \hspace{0.2 cm} (5)$ then $f(x) = x^{5} - 2$ has a unique solution in $\mathbb{F}_{p}$
To prove the statetament, i thought to define a linear application $$ \phi : \mathbb{F}_{p}^{*} \longmapsto \mathbb{F}_{p}^{*}$$
Define by : $f(x) = x^{5}$, studying the kernel of $\phi$ I noticed ...
0
votes
1answer
37 views
Why does not generator of multiplicative group generate all the members of group? [duplicate]
Generator(g) of multiplicative group (Zp where p is primary) is
an element the power of ...
1
vote
4answers
49 views
How to Show that a Field of Characteristic $0$ is Infinite [closed]
How can one prove that if a field $K$ has characteristic zero, then $K$ is infinite?
1
vote
1answer
25 views
A finite normal extension is also a splitting field
Let $E/K $ be a finite normal extension, then there exists $p(x)\in K[x] $ s.t. $E$ is the splitting field of $p(x) $.
The proof goes as follows:
$E=K(a_1,...,a_n) $ for some $a_1,...,a_n\in \Omega $ ...
0
votes
0answers
19 views
why separable algebra is a generalisation of separable field extension?
They are defined differently.
Suppose we have a field $K$.
We say a finite field extension $L$ is separable over $K$ iff the number of embeddings $L\hookrightarrow \bar K$ into the algebraic ...
2
votes
1answer
26 views
Example of a field of functions containing ln(x)
Up until now, I've mainly worked with the polynomial ring $\mathbb{R}[x_1,...,x_n]$ or the corresponding field of fractions. But I can't think of an example of a field of functions that contain non-...
2
votes
2answers
61 views
If a field $K$ (of characteristic 0) has no proper extensions of the form $K[\sqrt[n]{x}]$, is it algebraically closed?
Let $K$ be a field, assume characteristic 0 if this simplifies things. Suppose we know that for any positive integer $n$ and any $x \in K$ the polynomial $X^n - x$ has a root in $K$. Is $K$ ...
0
votes
2answers
60 views
Is there an efficient algorithm to find all zeros of systems of multivariate polynomial equations over a finite field?
I want my computer to solve large systems of multivariate polynomial equations over a finite field. The field is $\mathbb F_p$, where $p$ is a prime number. I heard that there is an algorithm using ...
2
votes
2answers
53 views
Is it 'rare' that $a$ and $a+1$ are conjugate (= have the same minimal polynomial)?
Let $a \in \bar{k}-k$, $k$ is a field of characteristic zero and $\bar{k}$ is an algebraic closure of $k$.
Denote the minimal polynomial of $a$ by $m_a=m_a(t) \in k[t]$.
Is it 'rare' that $m_a=m_{...
0
votes
0answers
30 views
If $a \in L-k $ satisfies $k(a^n)=L$ (for all $n \geq 1$), then $L/k$ is Galois?
Let $k \subsetneq L$ be a finite separable field extension, and let $a \in L-k$ satisfy: For every $n \geq 1$, $k(a^n)=L$.
In other words, all the non-zero powers of the primitive element $a$ are ...
4
votes
1answer
66 views
Galois group of $x^3-x^2-4$
In determining the Galois group of the polynomial $p(x) = x^3-x^2-4,$ I concluded that is must be the Klein-$4$ group as follows. First, $p(x) = (x-2)(x^2+x+2)$ and the roots of the irreducible ...
1
vote
0answers
51 views
$a,b \in \bar{k}$, such that $k(a)=k(b)=k(ab)$
Let $k$ be a field of characteristic zero, and let $a, b \in \bar{k}$ ($\bar{k}$ is an algebraic closure of $k$) be two distinct elements, such that
$k(a)=k(b)$.
Notice that $k(a)=k(b)$ implies that ...
-1
votes
0answers
21 views
Normal extension of fields
I have to prove that $\mathbb{Q}(\sqrt{2},\sqrt{3}, u$) is a normal extension over $\mathbb{Q}$, where $u^2=(9-5\sqrt{3})(2-\sqrt{2})$. Could someone help me to prove this? I claimed that $\mathbb{Q}(\...
0
votes
2answers
49 views
Degree of extension of fixed field by infinite set of automorphisms.
If $G$ is a finite group of automorphism $E \rightarrow E$, then Dedekind-Artin theorem tells us that $[E:E^G]=\; \mid G \mid$ where $E^G$ is the subfield of $E$ fixed by the automorphisms of $G$. Is ...
0
votes
0answers
18 views
Handling collisions by redundancy - Newton’s identities
I am reading this paper (P2P Mixing and Unlinkable Bitcoin Transactions
by Tim Ruffing, Pedro Moreno-Sanchez and Aniket Kate) and I need help in understand something in Section III-A, Handling ...
1
vote
1answer
22 views
Exercise about split closures (Galois Theory)
I am studing Galois Theory. I am using the books by Kaplansky, Fields and Rings. I am stuck doing this exercise:
Let $M$ be a split closure of $L$ over $K$ ($M,L,K$ are all fields). Prove that $M=L_1\...
0
votes
1answer
59 views
Suppose that $F$ is a field with $27$ elements. Show that for every element $a \in F$, $5a = −a$
Suppose that $F$ is a field with $27$ elements. Show that for every
element $a \in F$, $5a = −a$.
I am not able to understand how to approach this problem.
1
vote
1answer
25 views
Minimal polynomial for normal closure
I came across this problem while studying Normal closures.
Given $K=\mathbb{Q}$ and the polynomial $x^3-2\in K[x]$. $L/K$ is not normal where $L=\mathbb{Q}(2^{\frac13})$ since $\omega\not\in L $ ...
0
votes
0answers
43 views
Generator of $\mathbb{F}_{q}^*$ where $q$ is not prime.
I'm trying to find the generator of the multiplicative group $\mathbb{F}_{8}^*$, where $\mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would ...
4
votes
2answers
66 views
Why is $\mathbb{F}_{9}^*$ a multiplicative group
Since $\mathbb{F}_9$ is a field, its units $\mathbb{F}_{9}^* = (1,2,3,4,5,6,7,8)$ should form a multiplicative group. However in this group $3 \times 3 = 0 \notin \mathbb{F}_{9}^*$. I'm trying to ...
3
votes
0answers
44 views
About a problem of field extension in an algebra book by Fumiyuki Terada.
I am reading an algebra book by Fumiyuki Terada.
There is the following problem in this book:
$E_1, E_2, K$ are fields.
$K$ is a subfield of $E_1$.
$K$ is a subfield of $E_2$.
$p, q$ are ...
3
votes
1answer
67 views
Show a polynomial is irreducible over field $F(\alpha)$
In the field $F(u)$, let $\alpha = \frac{u^3}{(u+1)}$. Consider the subfield $F(\alpha)$ of $F(u)$.
Prove that the polynomial $f(x)=x^3-\alpha x-\alpha$ is irreducible over the field $F(\alpha)$.
...
