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 ...

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If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{{a_1}^p,\ldots,{a_n}^p \}$

Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent ...
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1answer
25 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If ...
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1answer
47 views

$G\subset G(F(x,y)/F)$ where $G=\langleσ\rangle$, $σ(x)=y$, $σ(y)=x$. Describe the subfield of $F(x,y)$ consisting of elements fixed by $σ$

Let $F$ be a field and let $E = F(x,y)$, where $x$ and $y$ are indeterminates. Let $G \subset G(E/F)$ be the subgroup defined by $G = \langle\sigma\rangle$, where $\sigma(x) = y$ and ...
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1answer
9 views

If $K/F$ is algebraic and $a\in K$ is separable over $F(a^p)$ then $a\in F(a^p)$

Suppose that $F$ is a field of characteristic $p$. Show that if $K/F$ is algebraic and $a\in K$ is separable over $F(a^p)$ then $a\in F(a^p)$. I know that the minimal polynomial of $a$ has ...
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0answers
33 views

What generates $\mathbb{F}_p^{\times}$ [on hold]

Say I have $\mathbb{F}_p^{\times}$ where $p$ is prime (this is the units modulo $p$). What would generate this field? EDIT: Is it isomorphic to the cyclic group $C_{p}$?
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41 views

When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
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1answer
40 views

The fixed field of $A$ is equal to the fixed field of $\langle A\rangle$.

Let $E$ be a finite extension field of $F$. Let $A$ be a subset of $Gal(E/F)$. Let $\langle A\rangle$ be the subgroup generated by $A$. Is the fixed field of $A$ equal to the fixed field of $\langle ...
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25 views

Is $D$ a field?

Problem. Let $D$ be an integral domain and let $\mathcal{F}(D)$ be a field of quotients of $D$. If $D\subset \mathcal{F}(D)$ then prove or disprove that, $D$ is a field. ...
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20 views

How are subgroups and subfields related in Galois theory [on hold]

My teacher for Galois theory said it was important not to mix up groups and fields within Galois theory Can someone explain how they are related and how to distinguish the Galois group and Galois ...
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0answers
17 views

If K is a field whose characteristic is not 2, show F/K is Galois [duplicate]

Let F/K be a field of extension 2, If K is a field whose characteristic is not 2, show F/K is Galois. I think I need to use a fact that the extension F/K is Galois if and only if K is the splitting ...
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0answers
15 views

Inseparable extension has zero discriminant

Let $K\subset L$ be a finite extension of fields. Show that if $L/K$ is not separable, then $\text{disc}(L/K) = 0.$ Edit: I will try to post the details of my work later, but here are the things I ...
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0answers
34 views

Splitting field of a set of polynomials

Given $X\subseteq F[x]$ where $F$ is a field, how to prove that there exist a splitting field of $X$ over $F$? In the case that $X$ is finite, I think the answer can be solved using Kronecker's ...
2
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1answer
36 views

Is there a systematic way of finding the factorization over the closure of $\mathbb{Z}_2$ of $p(x) = x^{32} - x$?

And similar polynomials of the form $x^{p^n} - x$. I know that the degrees of the irreducible monic polynomials that factorize $x^{32} - x$ will have degree $d \vert 5 = 1, 5$. Also, I know that $x$ ...
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1answer
34 views

Find basis in Extension field

I want to know that if we are asked to find the minimal polynomial, what are the steps? So if $F$ is a field and $\alpha$ is algebraic over F, first we need to find $[F(\alpha):F]$ and then ...
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0answers
14 views

CharK=0 (or p) iff CharF=0 (or p), F is subfield of K [duplicate]

Let $F$ be a subfield of the field $K$. Prove that: 1) $CharK=0 \iff CharF=0$ 2) $CharK=p \iff CharF=p,\ p$ is prime. My thoughts: (a) $1_K \in K$, so $ CharK=ord(1_K) \ | \ |K|$ from Lagrange. If ...
1
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1answer
20 views

Order of an orbit of Frobenius action on a algebraically closed field of characteristic p

Consider the action of the Frobenius homomorphism $F^{2}:\,\overline{\mathbb{F}_{q}}\rightarrow\overline{\mathbb{F}_{q}},\,x\rightarrow x^{q^{2}}$ over $\overline{\mathbb{F}_{q}}$ . Let $s=\left\{ ...
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0answers
22 views

Complex extension isomorphic to $\mathbb{R}$?

Let $K$ be some field extension of $\mathbb{Q}$ containing some complex number $c=a+bi$ with $a,b\in \mathbb{R}$ and $b\neq 0$. Is it possible that $K\cong \mathbb{R}$ as fields? I tried to disprove ...
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0answers
17 views

Problem when computing ideal class group

When computing the ideal class group of a quadratic extension $\mathbb{Q}[\alpha]$ after we have decomposed all rational primes smaller than the Minkowski bound into generating prime ideals ...
4
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0answers
30 views

The set of zeros of some polynomial over the closure of a finite field constitute a field in its own right

Let $p$ be a prime and $n$ be a positive integer and let $p(x) = x^{p^n} - x$ be a polynomial in $\mathbb{Z}_p[x]$. Let $Q$ be the set of all zeros of $p(x)$ over the algebraic closure of ...
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1answer
46 views

Find the minimum polynomial of $u$ over $Q$ where $u=\sqrt3-(1+(5/2)^{1/3})^{1/4}$ [on hold]

I tried using the binomial theorem but the terms keep increasing indefinitely
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1answer
16 views

Center of finite dimensional division $\mathbb{R}$-algebra?

Let $D$ be a finite dimensional division $\mathbb{R}$-algebra. Why is it that $Z(D)=\mathbb{R}$ or $Z(D)=\mathbb{C}$? I have seen an explanation: It is because $\mathbb{C}$ is the only non ...
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0answers
22 views

What does "class $x \in X$ mean?

I'm working through Escofier's book on Galois Theory, and in several exercises regarding finite fields they use the terminology "class $x \in X$, and I'm not certain what it means. For instance, let ...
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1answer
15 views

Problem on field extension related to irreducible polynomial

Suppose $\gamma,\gamma'\in\Bbb C$ are distinct roots of the same irreducible polynomial $p\in\Bbb Q[x]$. Suppose $x^2-5$ is irreducible in $\Bbb Q(\gamma)[x] $. Show that it is also irreducible in ...
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1answer
35 views

Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
0
votes
1answer
41 views

13th root of 2 in field $\mathbb{F}_{13}$

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
0
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1answer
40 views

Show that $\operatorname{Gal}(K/\mathbb Q)$ can be identified with the set of embeddings of $K$ into $\mathbb C$

I would be grateful if someone could help me demonstrate the following easy fact. Let $K$ be a number field which is Galois over $\mathbb Q$ and $\tau_0:K\hookrightarrow \mathbb C$ a fixed $\mathbb ...
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2answers
54 views

Isomorphic subfields of $\mathbb C$

Sorry if this is a very trivial question but I can't find a proof or a counterexample to it. If $K$ and $L$ are isomorphic subfields of $\mathbb C$ both containing $\mathbb Q$ then are they identical ...
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2answers
35 views

Computing the degree of the splitting field of $x^3+18x+3$ over $\Bbb Q$

Let $T$ be a splitting field of polynomial $$f(x)=x^3+18x+3\in\mathbb{Q}[x].$$ What is the degree $[T:\mathbb{Q}]$? My thoughts: the polynomial $f$ is irreducible over $\mathbb{Q}$, therefore the ...
1
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1answer
44 views

Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
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0answers
22 views

All possible degree 3 field extensions

Do we know anything about all possible degree 3 field extensions? If characteristic is 3: If Galois, then Artin-Schreier. If inseparable, then this is just cube root. If separable, but not normal, ...
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2answers
59 views

How to read this problem from Dummit-Foote's “Abstract Algebra”?

Problem 14.6.1 on page 617 says Show that a cubic with a multiple root has a linear factor. Is the same true for quartics? Let $f \in F[x]$ be a cubic. If $f$ has a root in $F$, let alone a ...
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0answers
23 views

Minimal polynomials over an inseparable extension

Let $F$ and $K$ be fields, and $\sigma$ and $\tau$ be two maps between them, $F \underset{\sigma}{\overset{\tau}{\rightrightarrows}} K$. Let $\alpha$ be an element algebraic over $F$, with minimal ...
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0answers
42 views

“Closure” of a polynomial ring by fraction field

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular noetherian $k$-algebra, $K$ the fraction field of $A$ and $\bar{K}$ an algebraic closure of $K$. Does there exist a ...
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2answers
26 views

Is there a way to see if $\alpha \in \mathbb{C}$ is constructible at a glance?

The notion of constructibility is not too obscure but mathematically, I find the definitions tedious and not very easy to handle with. I don't know if Ian Stewart's book Galois Theory edition 4 ...
2
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2answers
30 views

What is the definition of “prime ideal decomposition”?

I'm reading about Sunada's theorem in the book Geometry and Spectra of Compact Riemann Surfaces (Peter Buser) and I encountered this paragraph: If R is an algebraic number field and if $p \in ...
2
votes
3answers
41 views

$\mathbb{Z} [\sqrt{2}]$ is an integral domain

We know that $(\mathbb{Z} [\sqrt{2}],+,\cdot)$ is an integral domain. Someone can prove it easily if he says that is a subring of $(\mathbb{R} ,+,\cdot)$ . Can we find a different proof, more ...
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1answer
32 views

The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
3
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0answers
41 views

Kähler differentials in an inseparable field extension

Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. ...
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0answers
43 views

Prove that $\sin ^{-1} 1 $ is algebraic over $\mathbb Q$

Prove or disprove the following : $1.\sin ^{-1} 1 $ is algebraic over $\mathbb Q$ $2.\cos (\frac{\pi}{17})$ is algebraic over $\mathbb Q$ As suggested by @Andre ,for the 2nd one ...
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0answers
11 views

Algorithm for ordering on an algebraic number field

Given an algebraic field extension of the rationals $Q(P(X))$, where $P(X)$ is a polynomial in $X$, how do I algorithmically define an ordering on $Q(P(X))$ that is compatible with a specific real ...
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0answers
50 views

Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
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0answers
23 views

TO Find galois group of cubic pooynomial [closed]

$\text {prove that galois group of }x^3-4x+1\ \text{is}\ S_{3}$?? i have no idea how to approach this problem .please help
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1answer
46 views

If $x+y\sqrt{n} \in \mathbb{C}$ is a root of $f$ then $x-y\sqrt{n}$ is also a root

Let $n\in \mathbb{Z}$ be a non-square integer and $x+y\sqrt{n} \in \mathbb{C}$ a root of $f\in \mathbb{Q}[x]$ with $x,y\in \mathbb{Q}$. Show that $x-y\sqrt{n}$ is also a root of $f$. To show ...
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1answer
25 views

Algebraically closed field and polynomials

There is the problem: Let $F$ be a field of characteristic $0$ with the condition: If $f(x) \in F[x]$ has no roots in $F$, then the degree of $f(x)$ is a multiple of $21$. Prove that $F$ is ...
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38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
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2answers
30 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
2
votes
1answer
31 views

Additive combinatorics and $|S \cap (S+t)| \geq K |S|$ over $\mathbb{F}_2^n$

I don't know much about additive combinatorics, and I am wondering if there are results concerning the size of $|S \cap (S+t)|$ where $S \subset \mathbb{F}_2^n$ and $t \in \mathbb{F}_2^n$. Especially, ...
1
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0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
1
vote
0answers
39 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group ...
0
votes
0answers
33 views

Find the lattice of Galois Field

I am wondering what the lattice of subfield of $GF(p^{30})$ looks like. I know that it starts from $GF(p)$ and then $GF(p^2)$ and $GF(p^3)$, but then I am lost. And I looked it up online, but can't ...