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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
30 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
48 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
votes
1answer
38 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 ...
-2
votes
0answers
47 views

Computable Reals numbers property [on hold]

Definition: A real number $x \geq 0$ is computable if there are recursive functions $f$ and $g$ such that $g(n)$ has no zeros and $\lim_{n\rightarrow\infty} \frac{f(n)}{g(n)} = x.$ Show that the ...
0
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1answer
29 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 ...
0
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1answer
24 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 ...
0
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1answer
27 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 $F$?
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0answers
16 views

Why p-cyclic extension iff $p^m$-cyclic extension $\forall m$

A theorem is stated as follows. For a field $F$ of characteristic $p$, $F$ has a $p$-cyclic extension if and only if for every positive integer $m$, $F$ has a $p^m$-cyclic extension. I wonder if ...
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0answers
63 views

Showing equality of fields

I've come across some trouble in a proof and feel like I need a little push to reach the contradiction I'm aiming for! My goal is to show equality of the fields in the tower $K\subseteq K(a^{1/p})\...
3
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1answer
39 views

Is there a subfield of $\mathbb{R}$ that admits a non-Archimedean order?

We know any Archimedean ordered field can be embedded as a subfield of $\mathbb{R}$, but is there a subfield, on which there is another order instead of the one imposed by $\mathbb{R}$, such that it’s ...
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0answers
33 views

An example for ring with trivial ideals

I know that if there exists no ideal different from trivial ideals, the commutative ring with unity is a field. But I cannot figure out any example for a ring has trivial ideals only and not a field. ...
0
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0answers
16 views

algebraic field extensions and polynomials

Let $K$ be a field and $f$ be a polynomial, whose coefficients are algebraic over $K$. If we have a factorization $f=gh$ and the highest coefficient of $g$ is algebraic over $K$, is it true that the ...
1
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0answers
34 views

Duality between two multiplicative groups

Let $k$ be a field of characteristic $0$. Fix an integer $m$, and assume $k$ contains $m$ distinct roots of unity. Consider all finite extension of $k$ inside some fixed algebraic closure of $k$. (1)...
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0answers
9 views

Definition of exponent in Kummer extensions

I was looking Kummer extensions in the context of polynomials $x^n-a$ for $a\in F^*$, and splitting fields, minimal splitting fields, etc. I followed Lang's Algebra and Cohn's Basic Algebra. In both ...
2
votes
2answers
59 views

If $F\subseteq K$ are fields, $\alpha \in K$ Prove $\alpha$ is algebraic over $F$ [on hold]

If $F\subseteq K$ are fields, $\alpha \in K$, and $K$ is an extension field of $F$. Prove the following are equivalent: $\alpha$ is algebraic over $F$ $F(\alpha)=F[\alpha]$ $|F(\alpha):F|$ is ...
0
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1answer
27 views

If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Then $p(x) | f(x)$

If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Show: $$p(x) | f(x)$$ I'm not sure if it's a direct consequence of the proof that minimal polynomial of an ...
1
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0answers
20 views

What is the numerical norm for an Ideal?

I'm reading this text on Algebraic Number theory: (https://www.jmilne.org/math/CourseNotes/ANT.pdf, pg.68). They state the following: Let $\alpha$ be a nonzero ideal in the ring of integers $\...
3
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0answers
26 views

Relation between $[L \cap M : K \cap M]$ and $[L : K]$, and the Wantzel theorem

The well-known Gauß-Wantzel Theorem states that a real number $x$ can be constructed using straightedge and compass only if the minimal polynomial of $x$ (over the field $\mathbf Q$) has degree of ...
0
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1answer
29 views

Degrees of certain class of extensions of a field

Let $F$ be a field of characteristic $0$ and $\overline{F}$, its algebraic closure. Let $p$ be a prime number. Take $\alpha\in F^*$ and for this $\alpha$, choose $\beta\in\overline{F}$ such that $\...
1
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2answers
40 views

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 $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
0
votes
1answer
21 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
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0answers
51 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
votes
3answers
92 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
102 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
votes
1answer
31 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
32 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
77 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
70 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})/...
0
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0answers
39 views

Construction of Roots of Polynomials [closed]

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
28 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
23 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
27 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
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1answer
44 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
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1answer
18 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
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0answers
26 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
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1answer
33 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
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1answer
26 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
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1answer
36 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 $A$...
0
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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
51 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
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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
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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
43 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
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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 $abx^2$...
0
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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
42 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
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0answers
61 views

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

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?
2
votes
1answer
42 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 ...