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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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1answer
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Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/<x^3-x-1>$$ If $\theta$ is the image of x in $K$, then $\...
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1answer
20 views

$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
3
votes
0answers
45 views

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
8
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3answers
78 views

Showing that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$

I am attempting to show that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$, where $p > 2$ is prime. I have already shown that $[\mathbb{Q}(\sqrt[p]{2}, \sqrt{5}) : \mathbb{Q}] = 2p$. If needs ...
0
votes
2answers
98 views

How to prove $(-1)\cdot x = -x$ in a field? [duplicate]

I am having some hard time understanding the concept of field. I understand the 6 axioms: Associativity Distributivity Computability Closure Inverses Identities The exercise I am trying to solve ...
0
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1answer
28 views

Two questions on the ring S := {a+bs : a,b ∈ R}

Let $s$ be a formal symbol. Define addition and multiplication operations on the set $S := {a+bs : a,b ∈ R}$ (with curly brackets) by the rules $(a+bs)+(c+ds) := (a+c)+(b+d)s$, $(a+bs)(c+ds) := (ac+...
0
votes
0answers
29 views

Definition of the Norm Residue Symbol

Let $K$ be a field and $v$ its valuation. Let $K_v$ denotes the completion of $K$ with respect to the valuation $v$. If $L/K$ is a finite Galois extension, then all the $L_w$, for $w$ extending $v$, ...
2
votes
1answer
69 views

Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
2
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0answers
61 views

Field extension of degree $p^n$

I'm struggling with the following problem. Let n be a natural number, let $F$ be a field that contains a primitive $p^n$-th root of unity and let $a \in F^{\times}$. Show that if deg$(F(\sqrt[p]{a})/...
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0answers
39 views

Construction of Roots of Polynomials [on hold]

I'm wondering if I can construct a root of the following three polynomials: $x^2-7x-13$ $x^8-16$ $x^4+x^3-12x^2+7x-1$ I think I can because the field extension of a root and Q is always divisible by ...
0
votes
1answer
32 views

$A$ algebraic over $B$, $B$ algebraic over $C$. $A$ algebraic over $C$?

Let $A/B/C$ be field extensions, with $A$ algebraic over $B$, $B$ algebraic over $C$. Must $A$ be algebraic over $C$? I think the answer is yes, but I don't know how to prove it.
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0answers
26 views

Cardinalities of Galois extensions and compositum

Let $L_1/K$ and $L_2/K$ be Galois extensions contained in a (bigger) field $M$ and let $F$ be the smallest subfield of $M$ containing both $L_1$ and $L_2$ (in other words, $F$ is the field generated ...
0
votes
1answer
21 views

Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
0
votes
1answer
26 views

Is every point in the spectrum of a ring $R$ closed?

I am just getting started on spectrums of rings. I see how it is natural to augment the set of prime ideals with the Zariski topology, but from my poor intuition on the topic I don't see how any of ...
1
vote
1answer
42 views

Is there a ring whose total ring of fractions is not a field?

I am trying to come up with an example of a ring whose total quotient ring is not a field. I know that if $R$ is a domain, then every total quotient ring has to be a field, however in the general case ...
1
vote
1answer
16 views

Is a finitely generated torsion-free R-module free over R if R is an integral domain?

I know this is the case if $R$ is a PID, but PID's are special instances of Integral Domains, so I am wondering if there is a counter-example to the case where R is an integral domain. This post ...
0
votes
0answers
24 views

Must Additive Inverse elements be opposite in sign?

Is it possible to define a field $F=\left \{ 0, 1, a \right \}$ where the Additive Inverse condition is expressed as : $x+x=0 \space \space \forall x\in F$ ? My doubt comes from reading my book on ...
1
vote
1answer
32 views

Prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$.

For a Galois theory course, I need to prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$. Constructing the minimal polynomial ...
1
vote
1answer
25 views

Intermediate field and Galois extension.

I have the following problem: Let $p(x)=(x^{12}-16)(x^2-3)$. Show that $K=\mathbb{Q}(\sqrt[3]{2},\sqrt{3},i)$ is the splitting field of $p$ over $\mathbb{Q}$, $[K:\mathbb{Q}]=12$ and show that ...
1
vote
1answer
35 views

Is it possible for a finite integral closure of a DVR to not be a PID?

Suppose that we have a point $A$(local ring, DVR) of an abstract curve over $k=\bar{k}$ given by a field $k(X)$. Let $k(Y)$ be a finite extension of $k(X)$ and denote by $B$ the integral closure of $A$...
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votes
2answers
41 views

Is this structure a field?

I'm wondering whether the following addition and multiplication over the set $(\mathbb{R}\setminus\{0\}\times \mathbb{Z}) \cup \{0\}$ define a field: $$ (a,a')+(b,b')= \begin{cases} (a,a') \text{ if }...
2
votes
1answer
50 views

Fraction field and ring of integers

Let $K$ be a number field and let $O$ be a subring of the ring of integers $O_K$ of $K$. Show that $O$ contains a $\mathbb{Q}$-basis of $K$ if and only if the field of fractions of $O$ is $K$. I ...
1
vote
2answers
34 views

Splitting field of $x^3+x+1$ over $\mathbb{F}_2$ and the Galois group

I'm trying to find the splitting field $\Sigma$ of $f(x)=x^3+x+1$ over $\mathbb{F}_2$ and determine $\text{Gal}(\Sigma/\mathbb{F}_2)$. Please check if my reasoning is correct. Since $f$ is an ...
1
vote
1answer
33 views

Help $|nx|\le |n|\cdot|x|$, for $x\in K$ and $n \in \mathbb Z$?

How to prove: $|nx|\le |n|\cdot|x|$, for $x\in K$ and $n \in \mathbb Z$ ? The absolute value here is a nonnegative function from a field $K$ to $\mathbb R$ and in the definition there's a point; $|...
0
votes
0answers
40 views

Prove: Permutation of a root is another root of a polynomial

I read that Galois group is a permutation of the zeros or roots, this is new to me, so, I have a question. How can I prove, all roots of a polynomial are permutation of one another? in other words, ...
1
vote
1answer
37 views

$R=\mathbb{Q}[x]$ and $J=\langle x^2\rangle$, Is $R/J$ a principal ideal ring?

Let $R=\mathbb{Q}[x]$ and $J=\langle x^2\rangle$ Now I want to answer some question regarding $R/J$. First of all it is not a field. Also $ax, bx \in R/J$, both are non zero but their product $abx^2$...
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votes
0answers
26 views

L:K algebraic and separable then [L:K] $\leq$ n

Suppose that $L:K$ is an algebraic and separable extension, and that for every $\alpha \in L$, the minimal polynomial of $\alpha$ over $K$ has degree at most $n$. Then $[L:K]\leq n$. I'm having ...
0
votes
0answers
41 views

Affine $k$-domain of dimension$1$ can always be embedded in a polynomial ring?

Let $k$ be an algebraically closed field. Let $R$ be a UFD which is a finitely generated$k$-algebra. If $\dim R=1$, then is it true that there exists an injective $k$-algebra homomorphism from $R$ ...
-1
votes
1answer
41 views

Is the extension $\mathbb{Q}(\alpha)/\mathbb{Q}$, where $\alpha = 2\pi i /3$, a splitting extension? [closed]

Is the extension $\mathbb{Q(\alpha)}:\mathbb{Q}$ where $\alpha=e^{{2\pi i}/3}$ splitting extension because it has degree 2?
0
votes
0answers
59 views

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field [on hold]

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field? Because is not second degree. What is another way to show it, in this example?
1
vote
1answer
37 views

Can the real numbers be embedded into all non-Archimedean real closed fields?

Every Archimedean real closed field is isomorphic to a subfield of $\mathbb{R}$. But I’m wondering if something in the opposite direction is true. Suppose that $F$ is a non-Archimedean real closed ...
0
votes
2answers
44 views

Subfield and field extensions of a field with $27$ elements

I have constructed the field $\frac{\mathbb{Z}_3[x]}{<2x^3+x+2>}$ $=\{a_o+a_1x+a_2x+....+a_nx^n|a_i \in \mathbb{Z}_3, n \in \mathbb{N}, 2x^3+x+2=0\}$ $=\{a_0+a_1x+a_2x^2:a_i \in \mathbb{Z}_3\}$...
0
votes
0answers
9 views

Proposition about field of fractions [duplicate]

A proposition was left in abstract algebra course that is Fields of fractions of different integral domains are different too. I must prove if it’s true and give a counter example if it’s not true....
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votes
0answers
37 views

Understanding surjectivity of $G_{\mathfrak{P}}\to G(\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p}))$

I'm trying to understand Theorem I.9.4 from Neukirch's Algebraic Number Theory (page 56). First he proves that $\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p})$ is a normal extension, which is fine ...
3
votes
3answers
63 views

Let $\alpha$, $\beta$ be roots of $X^3-2$ and $X^3-5$ respectively, find $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]$

Given $\alpha,\beta \in \mathbb{C}$. Suppose $\alpha$ is a root of $X^3-2$ and $\beta$ is a root of $X^3-5$. Find the degrees $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]$. I know that $3=[\mathbb{Q}(\...
1
vote
1answer
50 views

Degree of splitting field of $f(x) \in \mathbb Q[x]$

I'm having trouble with a homework question, and I'm wondering if I can get some tips and/or hints (I do not want someone to solve the problem for me). The question is as follows: Let $f(x) \in \...
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votes
2answers
64 views

Find all roots of these two polynomials

There are only answers without any reasons in my textbook's questions. PLEASE HELP ME :( Find the root of these polynomials by using hints. Let the $w$ is complex root of $x^2+x+1$ 1.First ...
4
votes
0answers
89 views
+50

Proof of Fermat primes and constructible n-gon

Prove that if a regular n-gon is constructible, then $n = 2^kp_1 ···p_r $ where $p_1,...,p_r$ are distinct Fermat primes using the following facts. If the regular n-gon is constructible and n = qr, ...
2
votes
0answers
16 views

If a,b are separable and algebraic over an infinite field how do we show $\exists k \in K$ such that $K(a,b) = K(a+kb)$?

I want to know how to display this without any advanced Galois Theory or proving that there are finite intermediate fields.
1
vote
1answer
25 views

When are extensions of the polynomials with coefficients in the rationals isomorphic?

When is $\mathbb{Q}[x]/f(x) \cong \mathbb{Q}[x]/g(x)$, if $f(x),g(x) \in \mathbb{Q}$ are polynomials of degree 2? My initial approach to this question was to note that $\mathbb{Q}[x]/f(x) \cong \...
0
votes
0answers
23 views

Lattice and hyperplanes

I came across this problem and since I never worked with the lattice and hyperplane in the same context, I am not sure how to approach and solve this exercise: Let $L_n ⊆ R^3$ be the lattice of n^3 ...
1
vote
0answers
56 views

How to Determine Generators and Relations of Galois group of $X^7-2$

I try to understand how one may describe the Galois group of polynomials of the form $X^n-a\in\mathbb{Q}[X]$, in terms of generators and relations, where $X^n-a$ ($n>1$) is assumed to be ...
0
votes
2answers
24 views

Polynomial of $n+1$ Distinct Value and Uniqueness.

Let $F$ be a field and $f, g ∈ F[x]$ two polynomials of degree $n$ over $F$ . Suppose that there exist $n + 1$ distinct values $α_i ∈ F$ , such that $f(α_i) = g(α_i)$ for all $i$. How can I prove $f ...
0
votes
0answers
53 views

Trisect $\pi/5$

By writing π/15 in terms of π/5 and π/3, show that it is possible to trisect π/5 and also possible to construct a regular 15-gon. I'm not sure how breaking $\pi/15$ down like this would help me show ...
0
votes
0answers
23 views

Prove that $F(x)/F(\frac{x^3}{x + 1})$ is simple algebraic extension and find its minimal polynomial

Prove that $F(x)/F(\frac{x^3}{x + 1})$ is simple algebraic extension and find its minimal polynomial. I have not seen such tasks before, can you give me a hint or right algorithm to solve it, please?
0
votes
0answers
14 views

Find degree of generator element and defining equation

Find generator element and defining equation of this extension: $\mathbb{Q}(e^{\frac{2i\pi}{5}})/\mathbb{Q}$. Here is my try: Every element in $\mathbb{Q}(e^{\frac{2i\pi}{5}})$ can be presented ...
0
votes
1answer
27 views

Algebraic Number Subfield

Prove that the algebraic numbers A = {$x ∈ C|x\text{ is algebraic over }Q$} form a subfield of C. I'm not sure how to get started on this problem. I know what a subfield is but I don't know how to ...
3
votes
2answers
64 views

Outer Automorphism of $GL(n,F)$

Problem Let $\beta$ be a non-trivial aoutomorphism of the field $F$. Use $\beta$ to construct an outer automorphism of $GL(n,F)$. Do all outer automorphisms of $GL(n,F)$ arise in this way? My ...
0
votes
1answer
25 views

Galois field isomorphism [duplicate]

If α and β are roots of $x^3+x+1$ and$x^3+x^2+1$ ∈ $Z_2[x]$,respectively, prove that the Galois fields $Z_2(α)$ and $Z_2(β)$ are isomorphic. I have no idea how to solve this problem. I've gone ...
2
votes
1answer
35 views

Finding a Field in Which a Function Splits into Linear Factors

Let $f(x)=2x^3 +5x^2 +7x+6∈Q[x]$. Find a field, smaller than the complex numbers, in which f(x) splits into linear factors. I know that I should use the theorem that says that "If f(x) is any ...