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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Proof that all ordered fields are in the Surreals

It says on Wikipedia that any ordered field can be embedded in the Surreal number system. Is this true? How is it done, or if it is unknown (or unknowable) what is the proof that an embedding exists ...
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36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
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58 views

How the total order property of $\mathbb{R}$ is related to not being algebraicaly closed?

The field of real numbers $\mathbb{R}$ is total-ordered and not algebraicaly closed, the field of complex numbers $\mathbb{C}$ is not ordered but is algebraicaly closed. Intuitively how these two ...
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33 views

Is this proof that sigma is a fieldautomorphism legit?

I read a proof of the following theorem in "Basic abstract algebra" by Bhattacharya, Jain and Nagpaul and I thought that the proof looked overcomplicated. I have written my own proof of the theorem ...
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35 views

Is $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$ isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3)$?

Say we have the field extension $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$. Is this field isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3)$? I made some preliminary investigation, and this doesn't seem to be true. Is ...
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66 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
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31 views

Number of elements and number of different basis of $\mathbb F_5^3$

Let $\mathbb F:=\mathbb F_5$ the field with five elements. (i) How many elements has $\mathbb F^3$? (ii) How many different basis has $\mathbb F^3$? My idea: (i) $\mathbb F^3$ has $5^3$ elements. ...
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51 views

Is there a typo in this proof about Galois theory in Artin's Algebra?

The following is a statement in Artin's Algebra (2nd edition p. 489): Corollary 16.6.5 (a) Every finite extension $K/F$ is contained in a Galois extension. (b) If $K/F$ is a Galois extension, ...
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34 views

existence of purely inseparable extension

Let $F<E$ be a finite extension that is not separable. Show that for each $n\geq 1$, there exists a subfield $E_n$ of $E$ for which $E_n<E$ is purely inseparable and $[E:E_n]_i=p^n$ ($[...]_i$ ...
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42 views

Does $\operatorname{Proj}(\sigma)$ fix some points?

Consider a subfield $K\subseteq\mathbb C$, then by some properties of the fibered product of schemes we have that: $$\mathbb P^1_\mathbb C\cong\mathbb ...
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80 views

Proving that the algebraic reals form an infinite field extension over $\mathbb Q$

I believe I have proved the following, from the free textbook here, page 356. I have 2 different proofs and would greatly appreciate thoughts on them. Show that the set of all elements in ...
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30 views

Galois group of composite of Galois extensions

I'm reading through the proof in Dummit and Foote p. 593 that $$\operatorname{Gal}(K_1K_2/F) \cong H := \{(\sigma, \tau) \in \operatorname{Gal}(K_1/F) \times \operatorname{Gal}(K_2/F) \mid ...
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33 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
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27 views

Fields and quotient ring

Let $P(X)\in{\mathbb{R}[X]}$ irreducible polynomial. Then $\mathbb{R}[X]/(P(X)=X^2+1)\approx{\mathbb{C}}$. If $P(X)=X^2+X+1$ also $\mathbb{R}[X]/(P(X))\approx{\mathbb{C}}$? Or for a arbitrary ...
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56 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
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47 views

$\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism ...
2
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60 views

Factor polynomials into irreducibles over GF(q)

The polynomials $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$ are irreducibles over GF(2). (a) Factor both polynomials into irreducibles over GF(4). (b) Factor both polynomials into irreducibles over GF(8). I ...
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40 views

Field extension of complex root of cubic equation

If $c$ is a complex root of a cubic $a(x)\in\mathbb{Q}[x]$, show that $\mathbb{Q}(c)$ is the splitting field of $a(x)$ over $\mathbb{Q}$. For this, we must show that $\mathbb{Q}(c)$ contains all ...
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60 views

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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30 views

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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154 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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73 views

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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110 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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56 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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213 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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87 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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51 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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114 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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112 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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110 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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66 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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87 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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72 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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95 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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234 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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37 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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82 views

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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582 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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83 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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311 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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93 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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542 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
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277 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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104 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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40 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
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29 views

intersection of radical extensions of Q

Are there radical extensions $\mathbb{Q}\subseteq R_1$ and $\mathbb{Q}\subseteq R_2$ such that $R_1 \cap R_2$ is not radical over $\mathbb{Q}$? My guess is that one can find a such example, but I ...
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43 views

Prove that every sum of squares in $K$ is a square in $K$, where $K$ is certain field.

Let $K$ be a field such that $f(t)=t^{2}+1$ is an irreducible polynomial in $K[t]$. Let $i$ be a root of $f$ in an algebraic closure of $K$. Suppose every element of $K(i)$ is a square in $K(i)$. ...
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27 views

Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
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32 views

Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...