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.

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38 views

Multiplication operation on field with four elements whose underlying set is $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$?

Exercise from Aluffi, Alg: Chap. $0$ I know $\mathbb F_4 \cong (\mathbb Z/2\mathbb Z)[x]/(x^2+x+1)$. So, we must have $0 \leftrightarrow (0,0)$, $1 \leftrightarrow (1,1)$ and either $$x \...
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1answer
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$A$ is a field iff $A[t]$ is euclidean.

I'm almost sure the question has already been asked but i don't know the english terminologies... I have in my lecture that : $A$ a ring. $A$ is a field iff $A[t]$ is principal. I'm anoyed ...
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Definition of Algebraic Function Field

the definition of algebraic function field $\mathbf{F}$ over $\mathbf{K}$ that I know is the following: There exists an element $x \in \mathbf{F}$, transcendental over $\mathbf{K}$, such that $\...
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4answers
51 views

Clarification of proof in Artin's “Algebra”

I'm trying to understand a proof in Artin's Algebra "Theorem 15.7.3.b: the irreducible factors of the polynomial $x^{p^r}-x$ over the field $F_p$ are the irreducible polynomials in $F_p[x]$ whose ...
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Let $\alpha$ be algebraic over $\mathbb{Q}$ with $[\mathbb{Q}(\alpha) : \mathbb{Q}]=2$ and let $F=\mathbb{Q}(\alpha)$.

Let $\alpha$ be algebraic over $\mathbb{Q}$ with $[\mathbb{Q}(\alpha) : \mathbb{Q}]=2$ and let $F=\mathbb{Q}(\alpha)$. Suppose that $f(x) \in \mathbb{Q}[x]$ is irreducible of degree d. (i) If d is ...
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When is it possible to write an algebraic field extension as a union of a totally ordered (countable) family of finite subfields?

Let $L/K$ be an algebraic field extension. It is clear that $L$ can be written as a union of a directed family of finite subextensions, namely $$ L = \bigcup_{x \in L} K(x). $$ When is it possible to ...
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1answer
40 views

Galois Group of $\mathbb{Q}(\sqrt[3]{3},i\sqrt{3},\zeta_{13})/ \mathbb{Q}$

I would like find the Galois Group of $$\mathbb{Q}(\sqrt[3]{3},i\sqrt{3},\zeta_{13})/ \mathbb{Q},$$ that field extension is Galois because is the splitting field of separable polynomial $(x^3-3)(x^{...
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1answer
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How to see that $M$ is Galois over $k$ in Lang's proof that solvable extensions are a distinguished class? (Prop. VI.7.1, *Algebra*)

I am studying from Lang's Algebra, and in Chapter VI Galois Theory, $\S$7 Solvable and Radical Extensions, he states and proves the following proposition (on pages 291-292, third edition): ...
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Regular polygons with rational area

Problem: Find all natural numbers $n\ge 3$ such that the area of a regular $n$-gon of radius $1$ is rational. Given a circle of radius $1$, the only regular polygons inscribed in it with integer ...
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1answer
30 views

Let $E$ be an algebraic extension of $k$.

Let $\alpha$ be an element of $E$, let $p(X)$ be its irreducible polynomial over $k$, and let $E'$ be the subfield generated by all the roots of $p(X)$ which lie in E. Then $E'$ is a finite extension ...
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There is a formula for polynomial equations if and only if the generic polynomial is solvable

I'm trying to understand the proof that if $F$ is any field of characteristic $0$ there is no general formula for solving polynomial equations of degree $5$ or higher. I know the following facts: $1. ...
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41 views

Field extensions of odd degree

Problem: Let $L$ an extension of $K$ with $[L : K]$ odd. Show that $K(\alpha)=K\left(\alpha^{2}\right)$ for all $\alpha \in L \backslash K$ Solution:Let $L$ an extension of $K$ with $[L : K]$ odd. ...
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Do such field automorphism of $\mathbb{C}_p$ exists?

Let $\mathbb{C}_p$ be the p-adic complex field and $ \{a_i\}\subseteq \mathbb{C}_p$ be a finite set, then do there exist a field automorphism $\phi:\mathbb{C}_p\to\mathbb{C}_p$ such that for all $...
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1answer
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Discriminant not square free and monogenic

It is well known that if K is a number field whose discriminant is square free then K is monogenic. I want to know if the converse is true. If K is monogenic then is the discriminant necessarily ...
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48 views

Determine primitive element of constructible points of polynomial

I'm trying to determine the primitive element for the field $\Omega^{\text{constr}}\subset\Omega^f_{\mathbb{Q}}$ where $\Omega^f_{\mathbb{Q}}$ is the splitting field of the polynomial $$ f=x^9-2x^7+3x^...
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For which numbers $l \leq |K|$ field $K$ has subfield, which has $l$ elements? [duplicate]

char$(K) = p$, $|K|= p^n$, $|K^{\times}| = p^n-1.$ We know that $K^{\times}$ is cyclic group. Let $H$ be a cyclic subgroup of $K^{\times}$. Subgroup of cyclic group is also cyclic, so $H$ is cyclic ...
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1answer
49 views

A field extension, the larger field is algebraically closed, are there finite subextensions of arbitrary large degree?

Let $L/F$ be a field extension of infinite degree. Assume that $L$ is algebraically closed. I am not assuming that $L$ is an algebraic closure of $F$. Let $d\geq 1$. Must there be an intermediate ...
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2answers
46 views

What is the first order logic statement of “this field is of characteristic zero”?

I want to state that a field $F$ is of characteristic zero in logical notation to an audience without referring them to the meaning of the characteristic of a field. My first thought was the ...
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29 views

Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
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1answer
79 views

Galois group of degree 8 polynomial

What is the Galois group over rationals of the splitting field of the reducible polynomial $f(x)=x^8+14x^4+1=(x^4 - 2 x^3 + 2 x^2 + 2 x+1)(x^4 + 2 x^3 + 2 x^2 - 2 x + 1)=(x^4+2i\sqrt3 x^2+1)(x^4-2i\...
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Question about field of algebraic numbers [duplicate]

Suppose $\sigma \in$ Aut($\mathbb{A}$), where $\mathbb{A}$ is the field of algebraic numbers. Let $K = \{\alpha \in \mathbb{A} : \sigma(\alpha)=\alpha\}$. Let $E$ be a finite extension of $K$. ...
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1answer
53 views

Find the degree of a field extension

Suppose $F$ is the minimal subfield in $\mathbb{C}$ containing all the roots of polynomial $x^4-x^2+1$. Find the degree of a field extension $[F:\mathbb{Q}]$. I understand that I need to find the ...
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2answers
78 views

Roots of $f(x)$.

Let $p$ be a prime not equal to $2$. Let $f(x)$ be an irreducible polynomial over $\mathbb{Q}$ of degree $p$ with Galois group isomorphic to the dihedral group $D_{2p}$. I need to show that $f(x)$ has ...
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1answer
86 views

Show that the field extension is not Galois.

Let $\alpha$ be complex with $\alpha^2 = \sqrt{3} - \sqrt{5}$. I need to show that $\mathbb{Q}(\alpha)/\mathbb{Q}$ is not a Galois extension. Any hints would be very welcome.
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27 views

Extension of a field and $K$-place, Lemma 2.2.7 on Field Arithmetic

I'm studying field arithmetic on my own by reading the book "Field Arithmetic", third edition, by Michael D. Fried and Moshe Jarden. I have troubles to understand the proof of lemma 2.2.7. It says the ...
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2answers
33 views

Find all the middle fields of the extension $\mathbb{Q}/\mathbb{Q}(\exp({2\pi i\over 7}))$

The question: Let $F:=\mathbb{Q}, E:=\mathbb{Q}(\zeta)$ where $\zeta :=\exp({2\pi i\over 7})$. Find all the middle fields of the extension $E/F$. My attempt: The roots of $f(x):=\text{irr}(\zeta,...
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1answer
30 views

Normal basis of an extension of degree 4 over its prime field.

I want to construct a normal basis of $\mathbb{F}_{p^4}$ over $\mathbb{F}_p$, where $p$ is an odd prime. Is there any particular method to do it?
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36 views

Given a quadratic extension $K/F$, is there a quadratic extension $L/K$ such that $ \operatorname{Gal}(L'/F)=D_4$

Let $F$ be a number field (or a function field with constants $\mathbb{F}_q$ where $2 \nmid q$). If $L$ is some extension such that $[L:F]=4$ and $ \operatorname{Gal}(L'/F)=D_4$, the dihedral group ...
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42 views

Simple field extension in extension

Task: let $K$ be an extension of field $F$. Show, that $K$ include simple extension of field $F$. I denote $K^{'}$ as simple field extension of field $F$. Definition of simple field extension: if $\...
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1answer
36 views

A problem about prime subring

This is a problem from gtm 167 Field and Galois Theory. Let R be a commutative ring with identity.The prime subring of R is the intersection of all subrings of R.Show that this intersection is a ...
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1answer
26 views

Can the number of isomorphisms be less than the degree of field extension?

I am reading Nathan Jacobson's Basic Algebra I (2nd edition) in my private study. On p.227, there is a THEOREM 4.4. Let $\eta:a\to\bar{a}$ be an isomorphism of a field $F$ onto a field $\bar{F}$, $...
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1answer
59 views

Find roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}/(x^2+x+8)$

I need to find the roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}$ over field $\mathbb{Z}_{11}/(x^2+x+8)$ ($x^2+x+8$ is also primitive). As far as I'm concerned the answer is the ...
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0answers
17 views

Condition for normality of a simple extension, and question about $\operatorname{Aut}(K(\alpha)/K)$

My first question is this. If $K$ is any field and $\alpha$ is algebraic over $K$, I want to know if $K(\alpha)/K$ is normal if and only if $\pi_{\alpha,K}(X)$ splits in $K(\alpha)$ (where $\pi_{\...
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4answers
56 views

Explain why ($\mathbb{Z}_6$, +, · ) is not a field, where + is addition modulo 6 and · is multiplication modulo 6. [duplicate]

Explain why ($\mathbb{Z}_6$, +, · ) is not a field, where + is addition modulo 6 and · is multiplication modulo 6. When I was trying to explain why this is not a field, I came into a bit of trouble. ...
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0answers
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Why is the sign of the last term in a Sturm sequence always constant?

I have been given the definition of a Sturm sequence ($p_1,\dots, p_m$) as follows: $$p_0 = a \in \mathbb{Q}[x]\\ p_1 = a'\\ \forall 1 < i < m: p_i(t) = 0 \implies \text{sig}(p_{i-1}(t))=-\text{...
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0answers
31 views

The “reverse operation” of a field extension

It is possible to adjoin an element $\alpha$ to a field $\mathbb{F}$. Is there a name for the operation to take away an element $\alpha$ (and all subsequent elements) from a field $\mathbb{F}$ to form ...
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1answer
32 views

Norm in the cyclotomic field

Let $F = \mathbb{Q}(\xi_p)$ be the $p^{th}$ cyclotomic field. What is the norm of $N(1 + \xi_p)$? I’ve figured out that $N(1-\xi_p) = p$, as this can easily be seen from the minimal polynomial of $\...
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1answer
24 views

$\mathbf{Q}[2^{1/3}w]$ = $\mathbf{Q}[2^{1/3},w]$?

Consider $x^3 - 2$ with roots $2^{1/3}$, $2^{1/3}w$, $2^{1/3}w^{2}$ over $\mathbf{Q}$. where $\mathbf{Q}$ is the set of rationals and $w$ is the cube root of unity. Then which of the following are ...
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4answers
57 views

Prove $\mathbb Q[\sqrt[3]{2}]$={$a+b\cdot\sqrt[3]{2}$} is not a field [closed]

Suggestion: Prove that you can't write $\mathbb Q[\sqrt[3]{2}]$ like $a+b\cdot\sqrt[3]{2}$
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11 views

Transcendental Extension is Separable over Separable Closure

Let $K/k$ be a finitely generated field extension with transcendence basis $\alpha_1, \cdots \alpha_r$. I understand that this means $K$ is algebraic over $k':= k(\alpha_1, \cdots, \alpha_r)$ and $K$ ...
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0answers
30 views

Separable polynomials by Bourbaki

In Bourbaki, Algebra, V, 37, we have Proposition 3, which states that: Let $f$ be a non-zero polynomial in $K[X]$ and let $\Omega$ be an algebraically closed extension of $K$. The following ...
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2answers
60 views

Example of exotic $S_5$ as a Galois group

Is there an example of a sextic irreducible polynomial over $\mathbb{Q}$ with Galois group isomorphic to $S_5$? The transitive action of the Galois group of this polynomial on the 6 roots of $p(x)$ ...
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0answers
27 views

$p$'th roots in an extension of prime degree $p$.

Let $F$ be a field of characteristic $0$, and let $p$ be a prime number. Choose an element $r\in F$ and suppose that $r$ has no $p$'th root in $F$. Let $F':=F(\sqrt[p]{r})$, this is an extension of $F$...
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35 views

About the reduction modulo $p$ for calculating Galois groups

I know the following theorem: Let $f(x)\in\mathbb{Z}[x]$ be a monic separable polynomial of degree $n$. If $f(x)\mod p$ is separable and irreductible for some prime $p$, then there is a cycle of ...
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1answer
42 views

Prove that there exists a polynomial $p(x) \in \mathbb{Q}[x]$ of degree 21 such that $p(5^{1/3}+7^{1/7})=0$

Prove that there exists a polynomial $p(x) \in \mathbb{Q}[x]$ of degree 21 such that $p(5^{1/3}+7^{1/7})=0$ It would be nice if I had some simple theorem to gaurentee that $\mathbb{Q}(5^{1/3}+7^{1/7})...
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2answers
52 views

Is $x + x = 2x,\ x \in \mathbb{F}$ for all fields $\mathbb{F}$?

The question Obviously, a field has a $1$ and a $0$ element, the former being neutral regarding addition, the latter being neutral regarding the multiplication operator. Additionally, we know that in ...
3
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1answer
60 views

Is there any degree 6 irreducible polynomial in $\Bbb Q[x]$ whose Galois group is $A_4$?

I know that $A_4$ is a transitive subgroup of $S_6$, so transitivity is not critical in this problem. I tried to find irreducible polynomials of form $X^6-a(a \in \mathbb{Q})$, but their Galois ...
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2answers
49 views

Irreducibility of $x^n-a^n$ on $F(a^n)$ where $a$ is transcendental on $F$

I'm looking for a short proof of the following : If $F$ is a field where $a$ is transcendental, $x^n-a^n$ is irreducible on $F(a^n)$. I've managed to prove it in a tedious way but I feel like there ...
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1answer
31 views

Proof of finite subfields for a finite field extension

I just was looking at an exercise which asks the reader to show that for $F \subset L$, if $L = F(\theta)$ for some $\theta \in L$ then there exists only finitely many subfields $K$ of $L$ containing ...
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1answer
28 views

On isomorphism of a field of fractions

Let $F$ be a field, $\alpha \notin F$, and denote $F(\alpha)$ as a field of fractions that contain $F$ and $\alpha$. Is it true that $\frac{F({\alpha})}{(\alpha)}$ is isomorphic to $F$?