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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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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34 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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46 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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24 views

Splitting Fields Proof

Let $K\subseteq L$ be fields and $ f \in K[X] \setminus K$. Prove that the following are equivalent. (a) $L$ is a splitting field for $f$ over $K$ (b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ ...
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321 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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59 views

How does one characterise the reals without points?

The Real number line is defined upto unique isomorphism in the category $Fld$ as a Dedekind complete ordered field. One can view the Reals geometrically with its usual topology. In pointless ...
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39 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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34 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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77 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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73 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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56 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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62 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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38 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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352 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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106 views

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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82 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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89 views

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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123 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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65 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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328 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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67 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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157 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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143 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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131 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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72 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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94 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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75 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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100 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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315 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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39 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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94 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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751 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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87 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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383 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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77 views

Value groups of an archimedean field and its completion coincide

I am aware that for a non-archimedean field $K$ and its completion $\hat{K}$ with respect to a valuation $v$ (and corresponding absolute value $|\cdot|$), with $v$ extending to valuation $\hat{v}$ on ...
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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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310 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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Apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$

Question: I have to apply the Euclidean algorithm to $2$ and $1 − 3i$ in the integers of $\mathbb Q[\sqrt {-1}]$ My Solution: Since $N(2) = 4$ and $N(1–3i) = 10,$ we must start by dividing $1–3i$ ...
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24 views

Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
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10 views

Normality of towers of fields

If I have a tower of fields $F \subseteq E \subseteq K$ and K is finite and normal over F. I know that E needn't be normal over F because for example $\mathbb{Q}(\zeta_6, \sqrt{2})$ is normal over ...
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16 views

Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
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31 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
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20 views

What would be a value of $X$ under an automorphism of $F(X)$ over $F$?

Let $\sigma:F(X)\rightarrow F(X)$ be a field automorphism fixing $F$. What would be an value of $\sigma(X)$? Since $X$ is transcendental over $F$, $\sigma(X)$ is a transcendental over $F$ and ...
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24 views

prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$

$\mathbb{Q}<\mathbb{F}<\mathbb{C}$ - field extensions, such that $[\mathbb{F}:\mathbb{Q}]=m \in \mathbb{N}$ p is a prime number, $\alpha=p^{\frac{1}{n}}$ gcd(m,n)=1 prove n divides ...
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29 views

Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
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53 views

Cyclotomic polynomial, after adjoining a radical

Suppose that $p>2$ is prime and that $a$ is a rational number for which $\sqrt[p]{a}$ is in $\mathbb C\backslash\mathbb Q$. The cyclotomic polynomial $\Phi_p$ is well-known to be irreducible over ...
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23 views

Does $[F(\zeta_n) : F ]$ divide $\phi(n)$?

I know that if $F=\mathbb Q$, the degree actually equals $\phi(n)$. Also, if the extension $F(\zeta_n)/F$ is Galois, then I can invoke my knowledge of the existence of an injective map from ...