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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28 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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98 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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67 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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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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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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142 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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85 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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48 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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91 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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Hilbert’s zeros theorem, an application. (The algebraic variation)

Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
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102 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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103 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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61 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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81 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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69 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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93 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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225 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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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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475 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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290 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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92 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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105 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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499 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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259 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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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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31 views

Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a ...
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48 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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Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
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28 views

Elementary Field Theory: Extension Field of Degree 2

I'm trying to do/understand the following exercise: "Let $E$ be a finite extension of a field $F$. If $[E:F] = 2$, show that $E$ is a splitting field of $F$."* Background: Just beginning ...
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51 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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35 views

Computing degree of $a+b$ given $a$ and $b$ have coprime degree?

I'm reading a proof, but I'm not following the final conclusion. Suppose $a$ and $b$ are elements of degree $p$ and $q$ over a field $F$, where $p$ and $q$ are distinct primes. Then $[F(a,b):F]\leq ...
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35 views

A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$

Does this make sense as an alternative definition for a finitely-generated field extension?: A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism ...
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24 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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37 views

Composite of two simple extensions

Let $a, b$ be algebraic elements over a field $K$ and suppose at least one of these two elements is separable over $K$. Then, prove that there exists $c$ such that $K(a, b) = K(c)$.
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39 views

Factorize polynomial in $\mathbb Z_2[X]$

What is the most efficient way (less time consuming, algorithmically) to factorize polynomials in $\mathbb Z_2[X]$ ? For small degree polynomials, I just try every possibilities (like ...
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56 views

Separability of field extensions

I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra. $1)$ $L/k$ is ...
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34 views

kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
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52 views

Can “polar numbers” be added in a sensible way?

Let $\mathbb{P}$ denote the set of all "polar numbers," by which I just mean pairs of real numbers $(r,\theta).$ Note in particular that $r$ is allowed to be negative. Then we can structure ...
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39 views

Direct proof of: $\#($cover of $V) < \#F \; \Rightarrow \;V$ belongs to cover

I'm looking for a direct proof 1 (as opposed to a proof-by-contradiction) of the following theorem: Let $V$ be a vector space over a field $F$ and let $\mathcal{W}$ be a collection of (vector) ...
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36 views

A problem in field norm and division ring

let $D$ be a division ring with center $F$ and let $K$ be a maximal field of $D$. if $N_{D^*}(K*)$ be a maximal on $D^*$, then $K/F$ is galois.
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28 views

Extending field automorphisms

Consider an algebraically closed field $K$, and let $F$ be a subfield of $K$. Why every automorphism of $F$ can be extended to an automorphism of $K$? Notice that the field $K$ is not the algebraic ...
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29 views

Finding the smallest $k$ such that $f(x)$ divides $1-x^k$ where $f(x)$ is over $\mbox{GF}(2)$?

One technique is iterative that is to assume alpha as the root and solve for a higher exponent ($x$) until $\alpha^{x} = 1$. Is there any other technique?
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41 views

Cyclotomic polynomials to find the subgroups of a Galois group

With $f(x) = x^{10}+1$, I want to draw the lattice of subgroups of the group $Gal(L/\mathbb{Q})$. Using cyclotomic polynomials I find that we have the $Gal(\mathbb{Q}(e^{\frac{2 \pi i}{20}}) / ...
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17 views

Is there any vanishing criterion for elements of a tensor product of algebras?

Let $k \subset L$ be a field extension, $\mathrm{char}(k) = p$. I have some polynomial $f(X_1, \ldots, X_n) \in k^{1/p}[X_i], f^p \in k[X_i]$, with at least one coefficient not in $k$, such that ...
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47 views

Galois extension and subfield

Let $E/F$ be a Galois extension and $Gal(E/F)\cong \Bbb Z/p^3\Bbb Z$. Assume that there exists subfield $K$ of degree $p$ of $E$. (i.e, $[E:K]=p$) Then, show that any proper subfield of $E$ is ...
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46 views

Determing the structure of the subgroup of an automorphism group

Suppose we have two automorphisms of an extension field $L=\mathbb{Q}(t)$ for some variable, given by $\sigma: t \mapsto 1-t$ and $\tau : t \mapsto \frac{1}{t}$. Clearly $\langle \sigma , \tau ...