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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18
votes
0answers
326 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
192 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
10
votes
0answers
217 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
77 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
128 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 ...
8
votes
0answers
292 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 ...
7
votes
0answers
79 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
56 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 ...
6
votes
0answers
149 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 ...
5
votes
0answers
33 views

Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
5
votes
0answers
103 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
210 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
109 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
158 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
81 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
195 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
152 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
58 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
75 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
226 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
69 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
4
votes
0answers
102 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, ...
4
votes
0answers
51 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 $f(x)$ ...
4
votes
0answers
64 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
68 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
162 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
372 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
154 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 ...
4
votes
0answers
126 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
88 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
70 views

What is the Difference Between an Algebra and a Field?

The Wikipedia page for $\sigma$-algebra, as well as some other resources I'm studying, say this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on ...
3
votes
0answers
45 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
3
votes
0answers
38 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
3
votes
0answers
67 views

Why do fields seem to be a prerequisite for calculus?

I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation ...
3
votes
0answers
59 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
3
votes
0answers
166 views

About $Z_{p}[\sqrt{k}]$, when is it a field?

I give up. I'm new in the fields world, and I'm trying to give a sufficient and necessary condition for $\mathbb{Z}_{p}[\sqrt{k}]=\{a+b\sqrt{k}:a,b\in \mathbb{Z}_{p}\}$ to be a field ($p$ is a prime ...
3
votes
0answers
69 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 ...
3
votes
0answers
21 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
3
votes
0answers
53 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
46 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
50 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
64 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
79 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
140 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
82 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
55 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
87 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
119 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
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
82 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 ...