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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14
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277 views

When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$: ...
10
votes
0answers
196 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
9
votes
0answers
69 views

Squares in a Cyclotomic Extension

Is there an algorithm that can tell me whether some element of $\mathbb{Q}(\omega_n)$ is a square (i.e. in $(\mathbb{Q}(\omega_n)^{\times})^2$) ? An element $a = \sum_{i=0}^{\phi(n) - 1} a_i ...
8
votes
0answers
613 views

Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
7
votes
0answers
122 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
7
votes
0answers
78 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
6
votes
0answers
132 views

Limits and colimits in the category of fields

It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...
6
votes
0answers
258 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
5
votes
0answers
85 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
5
votes
0answers
126 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
5
votes
0answers
142 views

Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
5
votes
0answers
99 views

Irreducible polynomials of degree $d$ such that $[x]$ generates $\mathbf{F}_{p^d}^{\times}$

It is often convenient to represent the field $\mathbf{F}_{p^d}$ as $\mathbf{F}_p[x]/(f(x))$, where $f$ is irreducible with degree $d$, $f$ has just a few nonzero terms, and $[x]$ itself is the ...
5
votes
0answers
139 views

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$ Corollary $3.20$ page $267$ of Hungerford - Algebra: "Every proper ...
5
votes
0answers
48 views

When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let ...
5
votes
0answers
152 views

The Galois group of a composite of Galois extensions

Morandi's Field and Galois Theory, exercise 5.19b Let $K$ and $L$ be Galois extensions of $F$. The restriction of function map, namely, $\sigma\mapsto(\sigma\vert_K,\sigma\vert_L)$ induces an ...
5
votes
0answers
75 views

Is there a different copy of $\Bbb{R}$ in $\Bbb{C}$ such that the extension is algebraic?

Since $\Bbb{C}$ contains a non-trivial copy of itself, I know that there are multiple subfields of $\Bbb{C}$ isomorphic to $\Bbb{R}$. But these inclusions make the extension non-algebraic. So are ...
5
votes
0answers
185 views

Extension fields of $\mathbb Q$

Let $\mathbb Q$ be the field of rationals. Let $m_1, m_2,\dots, m_k$ be in $\mathbb N^*$. Let $t_1, t_2, \dots, t_k$ be in $\mathbb N$. Suppose $t_i^{1/m_i}$ $\neq $ $q t_j^{1/m_j}$ for ...
5
votes
0answers
138 views

Purely inseparable extension of algebraic function field

Let $K$ be a field with $\operatorname{char}(K) = p > 0$ and with the property that $[K:K^p] < \infty$. Let $F/K$ be an algebraic function field, i.e there is an element $x \in F$ which is ...
5
votes
0answers
55 views

Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= ...
5
votes
0answers
72 views

Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field

Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy to see that $[K : ...
5
votes
0answers
224 views

System of polynomial equations over rational field

Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations: ...
4
votes
0answers
62 views

Algebraical Fujita cancellation

If there exists some birational equivalence between ruled surfaces $C\times\mathbb{P}^1$ and $C'\times\mathbb{P}^1$ over a sufficiently nice field, then one can cancel the projective line and conclude ...
4
votes
0answers
66 views

Transcendental elements in $k[[x]]$ over the field $k(x)$

I have a hard time to prove that $k[[x]]$ contains an element which is transcendental over $k(x)$. Could you please explain the some idea how to do that?
4
votes
0answers
117 views

Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)

I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples ...
4
votes
0answers
304 views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
4
votes
0answers
113 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
4
votes
0answers
85 views

Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$

This is a follow-up question to this one. I asked it there first but moved it here following the advice from Cam McLeman. I tried to prove that $(\mathbb P:\mathbb Q)=\aleph_0$ and I think I ...
3
votes
0answers
22 views

How to lift field isomorphisms to splitting fields?

TL;DR: Proofs about arbitrary splitting fields are hard. Is there an easier way to do what I did? In our abstract algebra class, we were asked to prove the following theorem: Problem: Let $K$ ...
3
votes
0answers
44 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
3
votes
0answers
44 views

galois extension of imaginary quadratic field

Let $K$ be an imaginary quadratic field, and let $K \subset L$ be a Galois extension. Let $\tau$ denote complex conjugation. Show that $L$ is Galois over $\mathbb{Q}$ if and only if $\tau(L)=L$. My ...
3
votes
0answers
37 views

Splitting field containing $n$th root

Let $K$ be a splitting field of a polynomial over $\mathbb{Q}$. Suppose $K$ contains an $n$th root of some number $a$. Then how can we show that $K$ contains all the $n$th roots of unity? I don't ...
3
votes
0answers
92 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
3
votes
0answers
40 views

Multiplicative nature of the separability degree

In what follows, let $E / F$ be an algebraic extension, $h(x),f(x)\in F[x]$ polynomials, $h(x)$ irreducible. Definitions. We say $h(x)$ is separable if it have not repeated factors. We say $f(x)$ ...
3
votes
0answers
62 views

Factoring irreducible polynomial over normal extension

Let $f$ be an irreducible polynomial over $F$ and $K/F$ be a normal extension. How to prove $f$ is factored by product of irreducible poly. over $K$ with same degree? I tried to do it by if $f_1, ...
3
votes
0answers
73 views

Question about Wantzel's proof of the necessary condition for compass/straightedge constructibility

I'm trying to understand Wantzel's original proof of the necessary condition for constructibility with a straightedge and compass. It's expressed in terms of polynomials rather than field extensions. ...
3
votes
0answers
122 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
3
votes
0answers
78 views

Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see ...
3
votes
0answers
49 views

Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
3
votes
0answers
86 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
3
votes
0answers
110 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
3
votes
0answers
125 views

Determining whether or not an extension is a splitting field

On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest ...
3
votes
0answers
114 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
3
votes
0answers
74 views

The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
3
votes
0answers
128 views

Extensions Q(α,β) and Q(α∗β) over Q are the same one?? When?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha*\beta)$ over $\mathbb{Q}$ are the same one. Thanks in advance.
2
votes
0answers
36 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 ...
2
votes
0answers
19 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 ...
2
votes
0answers
23 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 ...
2
votes
0answers
38 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 ...
2
votes
0answers
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
votes
0answers
51 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 ...