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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95 views

Is $\Bbb Q[\sqrt2]$ cyclotomic?

This overview of Galois Theory claims that a field extension of $F$ is cyclotomic if it's obtained by adjoining an $n$th root of any element of $F$. Wikipedia claims you have to adjoin a root of unity ...
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387 views

Normal closure of field extension, axiom of choice

Update My previous proof was incorrect. This updated proof is inspired by the comment by 'MartianInvader'. Problem I can prove the statement 'Every algebraic extension $L:K$ has a normal closure ...
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1answer
88 views

proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions

Let $A$ be an integrally closed integral domain with field of fractions $K$ and let $p \in Spec(A)$. Let $L/K$ be a Galois extension with group $G$ and $L'/K$ be a finite Galois subextension. Let $B$ ...
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1answer
429 views

Algebraic closure of a field in the rational functional field is the field itself.

Let $k$ be a field. I need to show that the algebraic closure of $k$ in $k(x_1,x_2,...,x_n)$ is $k$, where the rational function field $k(x_1,x_2,...,x_n)$ is the field of fractions of the polynomial ...
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1answer
504 views

on the Galois Closure

In Brian Osserman's notes on infinite Galois theory, in the third paragraph of page 3 (proof of the fundamental theorem of Infinite Galois Theory) he says "let $E/F$" be THE Galois closure. I am not ...
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57 views

field extension $K\subset L$ and Uniqueness of $G$ such that $K=L^{G}$

Let $K\subseteq L$ be a algebraic field extension. If there exists subgroups $G_{1},G_{2}\subseteq$ Aut$(L)$ such that $K=L^{G_{1}}=L^{G_{2}}$, does $G_{1}=G_{2}$ holds? ($L^{G}=\{a\in L:\sigma a =a$ ...
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1answer
211 views

Regarding the normal closure of a finite extension

I was wondering how to find the normal closure of Q(∜2) in K, where K is the algebraic closure of the set of rationals. I think the normal closure should be Q(∜2,i),based on the fact that it is the ...
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1answer
63 views

prove that when $R$ is a ring and $a,b$ in $R$ satisfy $a + b = 0$, then $b + a = 0$ is also true

Prove that when $R$ is a ring and $a,b$ in $R$ satisfy $a + b = 0$, then $b + a = 0$ is also true. It seems apparently true but I don't know how to prove it.
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78 views

conjugate prime ideals of integral extensions and relevance of the characteristic of the ground field

This question refers to the proof of theorem 9.3, p. 66 in Matsumura's Commutative Ring Theory: "if $A$ is an integrally closed domain, $K$ its field of fractions and $L/K$ a normal field extension, ...
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96 views

Number field question

Show that $\sqrt3$ $\notin$ $\Bbb Q[\sqrt[4]2]$ The problem wishes us to use the trace function of a number field, and hints to write $\sqrt 3$ as $a+bx+bx^2+cx^3$. Why can we write it like this, ...
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198 views

Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?

It is not hard to prove that every element of $\mathbb{F}_p$ has a square root in $\mathbb{F}_{p^2}$: take any $a \in \mathbb{F}_p$ and consider the polynomial $f = X^2 - a$. If $f$ has a root in ...
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1answer
186 views

How can one show that no cyclotomic field contains $\sqrt[3]{2}$?

This is a paraphrase of a problem from an old exam that I'm going over. Show that for all positive integers $t$, when $\omega$ is a primitive $t$-th root of unity, $\sqrt[3]{2}$ does not lie in ...
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1answer
47 views

positive characteristic and multiple roots

I can't understand a proof in Milne, proposition 2.12 at pag 29. In particular, i can't prove the implication $c)\Rightarrow d)$ where: c) $F$ has characteristic $p\neq 0$ and $f$ is a polynomial in ...
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3answers
157 views

Is algebraic multiplicity of a root of a polynomial always smaller than the characteristic of a field?

I'm trying to prove a criteria for algebraic multiplicity of a root of a polynomial. Let $F$ be a field and $f\in F[X]\setminus\{0\}$. Let $r$ be the algebraic multiplicity of $c$ as a root of $f$. ...
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1answer
213 views

Can we make the set of all non-negative integers a field?

Can we define any kind of addition and multiplication on the set of all non-negative integers such that it becomes a field. I think not. Can we prove this ? If we have only a finite collection with ...
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1answer
102 views

How to find a field such that it is isomorphic to the set of formal fractions and power series?

Give examples of fields $F$ such that: $F$ is isomorphic to $F(t)$ $F$ is isomorphic to $F((t))$ $F(t)$ is isomorphic to $F((t))$. Thanks.
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3answers
291 views

Finding degree of $\mathbb{Q}(i, \sqrt{-1 + \sqrt{-3}}) : \mathbb{Q}$

Letting $\alpha = \sqrt{-1 + \sqrt{-3}}$, I have already that $|\mathbb{Q}(\alpha):\mathbb{Q}| = 4$ So one could do: $|\mathbb{Q}(i, \alpha):\mathbb{Q}| = |\mathbb{Q}(i, \alpha):\mathbb{Q}(\alpha)| ...
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2answers
124 views

Significance of the trace in isomorphic matrix fields

The field $\mathbb{Q}(\operatorname{i})$ has an isomorphic matrix field of degree two. The isomorphism is $$\varphi:x+\operatorname{i}\!y \longmapsto \left[\begin{array}{cc} x & -y \\ y & x ...
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1answer
91 views

Field norm defined with or without absolute value?

I'm studying about the valuation for Euclidean Domains and quadratic fields $\mathbb{Q}(\sqrt{\theta})=\{ \alpha: \alpha=a+b\sqrt{\theta}, a,b \in \mathbb{Q} \}$ and I'm not sure whether we start with ...
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1answer
57 views

Show that $q(\omega)\mapsto q(\omega^k)$ is an automorphism of $\mathbb Q$ given $\gcd(k,m)=1$.

Let $p(x)=x^m-1$ be a polynomial over $\mathbb Q$ and $E$ be the splitting field for $p$ over $\mathbb Q$. We know that $p$ has $\phi(m)$ primitive roots in $E$, where $\phi$ is the Euler's totient ...
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170 views

On the unicity of splitting field

I have this definition for the splitting field of a polynomial: let $f$ be a polynomial with coefficients in the field $F$. A field $E$ containing $F$ is called a splitting field for $f$ if it ...
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1answer
264 views

Galois group of $x^8+2$ over the rationals?

As to check whether I got the theory right I tried figuring out $ G=\Gamma(L:\mathbb{Q}) $ where $L$ is the splitting field of $f=x^8+2$ over the rationals. I then considered the intermediate field ...
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366 views

Square root in Characteristic 2 Field

Let $K$ be a field of characteristic 2. For each $a\in K$, can we always find some $x$ such that $x^2=a$? I came upon this question while reading "Arithmetic of Elliptic Curves". The original ...
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1answer
36 views

On equivalent definitions of minimal polynomial

Let $L|K$ be a field extension, let $\alpha$ be an element of $L$, algebraic over $K$. I want to show that the following definitions of minimal polynomial $f$ of $\alpha$ over $K$ are equivalent: 1) ...
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105 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
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2answers
801 views

Find all intermediate fields of extension $\mathbb{Q}(\sqrt{2} , \sqrt{3}) : \mathbb{Q}$ without using Galois correspondence.

I'm in the middle of some notes which claim it should be possible to show that all the intermediate fields of the extension $\mathbb{Q}(\sqrt{2} , \sqrt{3}) : \mathbb{Q}$ are - $\mathbb{Q}(\sqrt{2}, ...
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54 views

Condition for the number of distinct solutions over GF($q$)

Assume that we have $p$ sets $\left\{ {{m_i}} \right\}_{i = 1}^p$ with given cardinalities $\left\{ {{K_i}} \right\}_{i = 1}^p$, $1 \le {K_i} \le q$, where $q$ is a power of $2$. What I'm trying to do ...
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1answer
76 views

redundancy in definition of Galois extension?

In Artin's lecture notes on Galois Theory (Dover version), Theorem 15 page 44 says that "a field $E$ is a normal extension of field $F$ if and only if $E$ is the splitting field of a separable ...
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279 views

The simple extension $\mathbb Q(i+\sqrt{2})$

I want to a) show that $i$ and $\sqrt{2}$ are in $\mathbb Q(i+\sqrt{2})$ and that $[\mathbb Q(i+\sqrt{2}):\mathbb Q]=4$ b) find the minimal polynomial of $i+\sqrt{2}$ over $\mathbb Q$. ...
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114 views

Is $[\mathbb{R}:\mathbb{Q}]$ numerable?

Is there a numerable basis for $\mathbb{R}$ as v.s. over $\mathbb{Q}$? If $K$ is numerable field and $S$ is a numerable set. Is $K(S)$ numerable? I could only prove if $K$ is numerable and ...
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1answer
140 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
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57 views

Prove $ g \mid f $ and $ f \mid g \iff $ when $ f = c \cdot g $

Let $ f, g \in F[x] $, where $ F $ is a field. Prove that, $ g \mid f $ and $ f \mid g \iff $ when $ f = c \cdot g $ for $ c \in F^*$. I don't know nothing about it, so I please at help.
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155 views

Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
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1answer
228 views

Is it possible to do calculus on any field with a topology?

I'll try to make my point clear: when we consider the field of complex numbers $\mathbb{C}$ we can do calculus there because we have properties of a field and in the same time we have a topology to ...
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1answer
35 views

Number fields containing the values of a character

This is a basic question- Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ ...
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1answer
117 views

Irreducibility of Cyclotomic Polynomial over a Cyclotomic Extension

I am trying to prove that $\Phi_m(x)$ is irreducible over $\Bbb Q(\zeta_n)$ if and only if $(m,n)\leq2$. The left implication turns out to be somewhat easy since without loss of generality, $2\mid m$ ...
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1answer
207 views

Is $a^b$ transcendental when $a$ and $b$ are?

I am being asked this question as an exercise in Garling's "A course in Galois theory". But isn't this an open question in math?
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2answers
142 views

Ideal norm in a quadratic field

Let $K=\mathbb{Q}[\sqrt{d}]$ be a quadratic field with discriminant $d_K$, let $\mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2})$ be an ideal. Does the norm $N(\mathfrak{a})=a$? How to prove it?
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1answer
180 views

Determining A Splitting Field

I am trying to determine the splitting fields of a bunch of polynomials. I'll ask one here and hope that a general enough technique can be described to find the rest of them. Currently, I'm trying ...
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2answers
206 views

Algebraic Field Extensions

This is exercise $6.34$ from Lang's book: Give an example of a field $K$ which is of degree $2$ over two distinct subfields $E$ and $F$, respectively, but such that $K$ is not algebraic over ...
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246 views

Application of the Artin-Schreier Theorem

This is exercise $6.29$ out of Lang's book: Let $K$ be a cyclic extension of a field $F$, with Galois group $G$ generated by $\sigma$. Assume that the characteristic is $p$, and that ...
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200 views

how many elements are there in this field

$\mathbb Z_2[x]/\langle x^3+x^2+1\rangle $, I understand it is a field as $\langle x^3+x^2+1\rangle $ ideal is maximal ideal as the polynomial is irreducible over $Z_2$. but I want to know how many ...
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79 views

How can one go from “no roots” to “is irreducible” in this case? [duplicate]

This problem is paraphrased from an old version of an exam that I will be taking, and I have no idea how one would do solve it. Let $p$ be a prime number, let $F$ be a field of characteristic ...
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60 views

Minimal Polynomials of numbers from trigonometry

What is the minimal polynomial over $\Bbb Q[x]$ of $$\alpha[k] = n\cdot ...
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1answer
112 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
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1answer
108 views

Minimal Polynomial Trouble

Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an ...
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744 views

What is the field $\mathbb{Q}(\pi)$?

I'm having a hard time understanding section 29,30,31 of Fraleigh. In 29.16 example, what is the field $\mathbb{Q}(\pi)$? and why is it isomorphic to the field $\mathbb{Q}(x)$ of rational functions ...
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1answer
98 views

Element of with n0n-zero trace

Let $F$ be a field of characteristic $p$ and $K$ a finite, separable extension of $F$ such that $p \mid [K : F]$. I want to show that there must exist an element of $K$ with non-zero trace. One idea ...
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129 views

Can I embed $\Bbb{C}(x)$ into $\Bbb{C}$?

Is it possible to embed $\Bbb{C}(x)$ (the field of rational functions over the compelx numbers) in $\Bbb{C}$ ?
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216 views

Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$

Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$. Could you help me prove this theorem? My professor introduced this ...