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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Relationship between the minimal polynomial of $\sin{2^{\circ}}$ and $\sin{5^{\circ}}$ over $\mathbb Q$

Let $f(x)$ be the minimal polynomial of $\sin{2^{\circ}}$ over $\mathbb Q$, and $g(x)$ be the minimal polynomial of $\sin{5^{\circ}}$ over $\mathbb Q$, then $f(x)+f(-x)=2 g(x)\tag 1$. I find this ...
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Proving integrality of the coefficients “inside the box”

Consider the (usual) $ABKL$ setting: $A$ is an integral domain with field of fractions $K$, $L/K$ is an algebraic field extension, and $B$ is the integral closure of $A$ in $L$ (we are not assuming ...
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267 views

The automorphism group of a field with $p^2$ elements

Suppose $K$ is a field. Then we call $f: K\to K$ a (field) automorphism if $f$ is a one-to-one, onto and unital (i.e. $f(1)=1$) homomorphism of rings. The following results are well-known. There ...
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Automorphism of Field Extension Statement

In our Galois Theory course the following statement was given to us in lectures: Suppose that $L : K$ is finite and normal, and $α, β ∈ L$ are roots of an irreducible polynomial $m ∈ K[x]$. Then ...
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Proving Lagrange's Theorem with Galois Theory

Problem: Let $G$ be a finite group with subgroup $H$. Let $|G| = n$ and $|H| = m$. Prove that the order of $H$ divides the order of $G$ using only results from Field and Galois Theory, with the ...
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embedding theorem in field theory

Let $C$ be an algebraic closure of $E$. Let $E/F$ be a finite separable extension of degree $n$, and let $\sigma$ be an embedding of $F$ into $C$. Then $\sigma $ extends to exactly $n$ embeddings of ...
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120 views

Another galois theory problem. (quartic with dihedral Galois group)

Given: If $f$ in $k[X]$ is an irreducible quartic and $G$ is the galois group of its splitting field, then $G$ is contained in $D_8$ iff the resolvent cubic of $f$ has a root in $k$. Now suppose ...
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64 views

On cyclic extensions.

I would really appreciate it if you help me with the following problem: Suppose K is algebraically closed, f is an automorphism of K of in finite order, and k is the elements of K fixed by f. Show ...
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278 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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92 views

Necessary criterion for a field extension to be normal

I'm working on a lemma concerning some Galois theory and arithmetics. Let $p$ be an odd prime and $K/F$ be a finite Galois extension of number fields of order prime to $p$ with Galois group $H$. Let ...
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61 views

Field extension

I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$. I know I have to prove that $ug(x)-f(x)$ is ...
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135 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
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139 views

Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
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123 views

Fixed points of automorphism in the field $\mathbb{C}(x,y)$

I am trying to solve a problem, and one of the parts is the following: let $M=\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ be a non singular $2\times 2$ matrix with integer ...
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69 views

Algorithm for determining whether two polynomials have the same splitting field

This question asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. ...
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92 views

Theorem's relying on algebraic closures

When working with fields, it's a usual method to work on an algebraic closure of a field to obtain results about that field. In general (i. e. unless you're explicitly considering "well-behaved" ...
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73 views

Structure of $\mathbb{Q}_p(\zeta_p)$

Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in ...
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98 views

Conditions for the degree of field extension reaches its upper bound

(Re-edited) Let $k$ be a field and $a,b$ algebraic over $k$ but not inside $k$ and $a \neq b$. Suppose that $[k(a,b) : k]=[k(a):k] [k(b):k]$. What does this tell us about the relation between $a,b$? ...
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289 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
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38 views

Real embeddings and linear disjointness

Suppose I have two Galois extensions $F_1, F_2$ of $\mathbb{Q}$ such that $F_1$ has a real embedding. Then is there a general condition on $F_2$ such that $F_1$ and $F_2$ will be linearly disjoint ...
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Given a normal basis $\{g(a)| g\in Gal(L/K)\}$ of $L/K$ finite Galois, can we express $g(a)g'(a)$ relative to this basis?

Let $L/K$ be a finite Galois extension. The normal basis theorem says that there is an element $a\in L$ such that $\{ g(a) | g\in \text{Gal}(L/K)\}$ is a basis of $L$ as a $K$-vector space. Let ...
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684 views

A proof of the normal basis theorem of a cyclic extension field

I came up with the following proof of the normal basis theorem of a cyclic extension field. Is this proof well-known? Proposition Let $L$ be a finite cyclic extension of a field $K$. Let $n$ be the ...
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86 views

Intersections of finitely generated field extensions are finite?

I was reading the following post at MathOverflow: http://mathoverflow.net/questions/21086/when-are-intersections-of-finitely-generated-field-extensions-finitely-generated/21093 I can't comment there, ...
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337 views

Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...
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94 views

defining binary operation on $\mathbb{R}$

Exercise 3, page 13 from Golan's book ("The Linear Algebra a Beginning Graduate Student Ought to Know"): Define a new operation $\circ$ on $\mathbb{R}$ by setting $a\circ b= a^{3}b.$ Show that ...
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108 views

Quick question: finite extensions and norms

[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."] I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
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294 views

field embedding

I've come across this problem in Etingof's notes on representation theory (Problem 5.1 on p. 78). It just sounds nice exercise... The question is : Let $f : k(x_1,\ldots,x_n)\rightarrow ...
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107 views

Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
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The most general splitting of a field extension

Take $L/K$ an extension of the field $K$. I have questions on how we can "split" the extension $L/K$. (1) One knows that $L$ has a transcendence basis $(x_i)_{i\in I}$ over $L$, so that if $E := ...
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27 views

Mistake in proof that a polynomial $f$ irreducible in $F$ is irreducible in $E$ if $\gcd(\deg f, [E:F])=1$

This is a problem in James Milne's text on Galois Theory: Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F] = m$. If $\gcd(m,n)= ...
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21 views

Extension Fields, complex numbers.

I have a question about the complex numbers given as an extension field. I know that complex numbers can be seen as $\mathbb{R}\left[x\right]/<x^2+1>$ (Well, I really don't know it yet, because ...
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The equation has a solution iff $(k,q-1)=1$

Let $p^n$, where $p$ is a prime, and $k \in \mathbb{N}$. Let $a$ the generator of the cyclic group $\mathbb{F}_p^{\star}$, where $\mathbb{F}_q$ the finite field with $q=p^n$ elements. Show that the ...
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24 views

Orbits of elements of the field under the action of a subgroup of field automorphisms

Let $G$ be a subgroup of the group of automorphisms of field $E$, $\mid G\mid = n$. Can we find such $\alpha\in E$ that its orbit under $G$ consists of $n$ different elements? If $E$ is a Galois ...
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27 views

Field of polynomials mod n?

I have a few questions and i am looking for some clarification. 1) Is it correct that one can define a field $(Z_n, +, X)$ of integers mod $n$, where all the elements are integers $a$ such that ...
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45 views

How to show that $[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9$

How can I show that $$[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9?$$ My idea is: $\sqrt[3]2$ has for its minimal polynpmial $X^3-2$ over $\mathbb Q(i\sqrt 3)$, which I justify by: ...
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Finding a Galois group using a cubic resolvent

I've heard there's a fast way to find the Galois group of a quartic polynomial using its resolvent. Can anyone explain how it's done or give a reference? Is this method the same as the general ...
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49 views

The number of automorphisms of a finite field

Let $M$ be a finite field and $|M| = p^s$, where $p$ is prime and $s \in \mathbb N$. Prove that the number of different isomorphisms field $M$ to $M$ equal to $s$ and this isomorphisms form a cyclic ...
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44 views

Irreducibility of a polynomial with algebraically independent coefficients

I am learning some kind of field theory. Let $\mathbb{Q}'$ be the smallest subfield in $\mathbb{C}$ containing all roots of unity. Recently I read a book on Galois theory and met the following ...
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55 views

If $F$ is finite over $L_1$ and $L_2$, is it also finite over $L_1 \cap L_2$?

Let $F$ be a field and $L_1$, $L_2$ two subfields such that $F$ is finite over both $L_1$ and $L_2$. Is $F$ necessarily finite over the intersection $L_1 \cap L_2$?
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25 views

Euler's criterion and Legendre symbol

I am working on an exercise which is the following : Let be $n$ an odd integer and $b$ such as $b \wedge n = 1$, then $(\frac{b}{n}) \equiv b^{(n-1)/2} \mod n$. (*) If $n$ is divisible by the ...
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44 views

$x^k = na$ has solution over the field $F, \text{char} F \neq 2$

I came up with an interesting Hypothesis. Suppose we are in the field $F, \text{char }F \neq 2$. Let's fix an arbitrary element $a \in F$. Is it true that at least one equation $x^k=n\cdot a$, ...
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Verification of proof that the left distributive property holds on a field of quotients, F

Here's my proof. I would like to know whether it is right or wrong and if the latter where it needs to be corrected: (a,b)=[(a,b)][(cf+ed,df)]=[(acf+aed,bdf)]. We want to show its equivalence to: ...
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55 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
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27 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
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21 views

Field extensions and quotient fields

STATEMENT: Suppose that $F\subseteq E$ is a field extension of $F$. And assume $u\in E$ is transcendental over $E$. Then it readily follows that $F(u)\cong F(x)$, where $F(x)$ is the quotient field of ...
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45 views

P. Morandi on p-closure of a field

I am stuck on a step of the proof of Lemma 18.4 of Patrick Morandi, Field and Galois Theory. Let $p$ be a prime number and let $F$ be a field with $\mbox{char}(F) \neq p$. Morandi defines the ...
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47 views

Examples of a complete ordered field

We know that every complete ordered field is isomorphic to $\mathbb R$, but are there examples of complete ordered fields different, not isomorphically different of course, from $\mathbb R$?
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19 views

Is $PE$ is purely inseparable over $E$?

I want to prove the statement : if $K\le E\le F$ and $F$ is separable over $E$, then $P\subset E$. Here, $P=\{u\in F:\text{$u$ is purely inseparable over $K$}\}\le F$. I claim that $PE$ is both ...
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44 views

Diagrams characterizing ring characteristic, and in particular field characteristic 1?

For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$? Is there a diagram, or the shape of what ...
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31 views

Algebraic extension of rational functions

Let $k\subset F\subseteq k(X)$ be chains of field extension, prove that $k(X)/F$ is algebraic. "Proof:" Let $y\in F\setminus k$ then $y=\frac{P(X)}{Q(X)}$ with $P\notin k$ or $Q\notin k$. It ...