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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6
votes
2answers
109 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
1
vote
1answer
29 views

Is $\mathbb{F}_3(t)(\sqrt[4]t)$ a splitting field over $\mathbb{F}_3(t)$?

Let $t$ be transcendental over $\mathbb{F}_3$. I used the polynomial $f = x^3 - 1$ to prove that $\mathbb{F}_3(t)(\sqrt[3]t)$ a splitting field over $\mathbb{F}_3(t)$. This was convenient since $f = ...
0
votes
1answer
173 views

Is this right? Proof that the field of rational numbers is not isomorphism to rational forms over rationals

A rational form is defined like this: Let $F$ be any field. Write $F[t]$ for the set of all polynomials $p(t)=a_0+a_1t+\cdots+a_nt^n$ in an indeterminate $t$, with coefficients $a_k$ in $F$. ...
1
vote
0answers
77 views

Field extension of fraction field of polynomial ring modulo an ideal.

My apologies for the relatively long question, but I am trying to understand a step in a proof, which needs some preliminary explanation. Let $K$ be a field and $I$ a prime ideal of ...
3
votes
2answers
54 views

Splitting field of $f$ as smallest field extension containing all BUT ONE zero of $f$

I'm just working with splitting fields and I have to prove something which I don't understand. Let $L$ be a splitting field of the polynomial $f$ over $K$ and $f = \prod_{i=1}^n(X-\alpha_i)$. ...
2
votes
0answers
76 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 ...
2
votes
1answer
246 views

Decomposition of tensor product into direct sum of fields

If I have tensor product of two fields $V_1\otimes V_2$, what is the general approach to decompose this product into a direct sum of fields? In particular, I have $\bullet\;\Bbb Q(\sqrt 2) ...
1
vote
3answers
205 views

Questions about $\Bbb Q[\sqrt{p}]$ and $\Bbb Q(\sqrt{p})$

I studied this part where they talk about $\Bbb{Q}(\sqrt{2})$ and $\Bbb{Q}[\sqrt{2}]$ and I really start to get confused. Definitions: $$ \Bbb{Q}[\sqrt{2}] = \left\{ a + b \sqrt{2} \mid a,b \in ...
0
votes
0answers
22 views

If $\dim_K F<\infty$, must there exist $t\in F$ such that $K[t]=F$

Let $F$ be a finite dimensional extension field of a field $K$. Must there exist $t\in F$ such that $K[t]=F$? I think I can prove this when $\operatorname{Char}K=0$, but what about the other cases? ...
2
votes
2answers
39 views

Does $(x-a)^n\in K[x]$ imply that $a\in K$?

Let $F$ be an extension field of $K$. Let $a\in F$ and $n$ be a positive integer. It is also given that the polynomial $(x-a)^n$ has all of its coefficients in $K$, i.e. $(x-a)^n\in K[x]$. Does it ...
1
vote
2answers
347 views

Why does the minimal polynomial of α divide all polynomials for which α is a root?

Suppose α ∈ E with E being a field extension of F. Let S be the set of all polynomials in F[x] for which α is a root. Prove that the minimal polynomial of α over F divides every polynomial in S.
2
votes
1answer
136 views

An example concerning some fields

I was trying to understand the following example This is also why I've made some questions today on locally finite fields. I've almost understand everything (thanks also to some of you) but I ...
1
vote
1answer
70 views

Projective special linear group

What is it the minimum number of generators for $PSL(2,\, \mathbb{F}_q)$? Is it known? Is there some references I could see?
1
vote
1answer
84 views

Locally finite infinite field

Is there a place (a book maybe) where I can find some useful information on infinite locally finite fields? Especially when all of whose proper subfields are finite? I know, for instance, that a ...
2
votes
1answer
52 views

Separability of a polynomial

I have a non zero polynomial $f\in F[X]$ where $F$ is a field. Let $L$ be a field extension of $F$ so that $f$ splits completely in $L[X]$, so $f(X)=c\prod_{i=1}^n (X-a_i)$ with $c,a_i\in L$. If ...
1
vote
0answers
50 views

Showing that $\frac{\mathbb{C}[X]}{<x-1>}$ is isomorphic to $\mathbb{C}$

I'm trying to show that $\frac{\mathbb{C}[X]}{<x-1>} \cong \mathbb{C}$ and I am not sure if this argument is correct. Define $\phi: \mathbb{C}[X] \to \mathbb{C}$ by $\sum a_it^i \to \sum a_i$. ...
1
vote
4answers
56 views

Proving $\mathbb{Q}(\eta,i)=\mathbb{Q}(\xi)$ where $\xi=e^{\pi i/10}$ and $\eta=e^{2\pi i/5}$

Proving $\mathbb{Q}(\eta,i)=\mathbb{Q}(\xi)$ where $\xi=e^{\pi i/10}$ and $\eta=e^{2\pi i/5}$. All I have got to prove is $[\mathbb{Q}(\eta):\mathbb{Q}]=4$, $[\mathbb{Q}(\xi):\mathbb{Q}]=8$ and ...
2
votes
0answers
73 views

Did I Do This Galois Theory Problem Right? Subfields of $\mathbb{Q}(\zeta_{12})$.

Let $\omega$ be a primitive $12$th root of unity. (i) What is $[ \mathbb{Q}(\omega) : \mathbb{Q}]?$ (ii) List the distinct conjugates of $\omega + \omega^{-1}$. (iii) What is $Aut(\mathbb{Q}(\omega ...
1
vote
2answers
90 views

Is my answer correct on this Galois Theory problem? Find the lattice of subfields of $\mathbb{Q}(\zeta_9)$

Problem: let $\zeta$ be a primitive $9$th root of unity, and $K = \mathbb{Q}(\zeta)$. Describe the lattice of subfields of $K$, give generators for each subfield and list its degree over ...
5
votes
0answers
57 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 ...
0
votes
3answers
42 views

need help explaing the complex roots of a cubic

I am trying to understand a Galois theory example and we are looking at the solutions of $x^3-2=0$. It says they are $2^\frac{1}{3},2^\frac{1}{3}\omega, \text{ and } 2^\frac{1}{3}\omega^2$. I know ...
3
votes
3answers
75 views

Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
0
votes
0answers
134 views

Splitting fields and isomorphic

Please check these statements whether those are true Let $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ be extension fields of $\mathbb{Q}$, the rational numbers. Since they are not isomorphic as ...
0
votes
0answers
54 views

Dummit & Foote: Construction of the regular $17$-gon

In the last step of the exercise where we construct the $17$-gon, I have to draw a circle with a diameter whose endpoints are $(0, 1)$ and $(\eta_1', \eta_4')$ and show that it intersects the positive ...
3
votes
0answers
88 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))= ...
0
votes
1answer
64 views

Help Understanding Field Extension/Linear Algebra Problem

Let $E = F(\alpha)$ be a simple field extension of a finite field F by an algebraic element $\alpha$. Thinking of $E$ as an $F$-vector space, define a linear transformation $$T:E\rightarrow ...
3
votes
1answer
104 views

Question about the number of roots of irreducible polinomials.

Suppose $F|K$ is a field extension and let $\alpha \in F$ be an algebraic element over $K$ and let $\beta \in K(\alpha)$. Let $f=Irr(\alpha, K)$, $g=Irr(\beta, K)$, $h=Irr(\alpha, K(\beta))$. Let $L$ ...
1
vote
1answer
110 views

Let $F$ be a field of 8 elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number k<1}. Then the number of elements in A is

Let $F$ be a field of $8$ elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number $k<1$}. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 Please give me some ...
3
votes
2answers
274 views

Field homomorphism into itself

Suppose $F$ is algebraic over $\mathbb{Q}$ and $\varphi \colon F \to F$ is a homomorphism. Prove that $\varphi$ is an isomorphism. I came across this question here but I don't quite get why $F$ ...
3
votes
1answer
194 views

Minimum polynomial of $\sqrt{2} + \sqrt[3]{5}$ above $\mathbb{Q}$ (and a generalization)

I have found that $\alpha = \sqrt{2} + \sqrt[3]{5}$ is a root of $f(x) = x^6-6x^4-10x^3+12x^2-60x+17$. I don't know if this is the minimum polynomial of $\sqrt{2} + \sqrt[3]{5}$ above $\mathbb{Q}$. I ...
5
votes
2answers
798 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
1
vote
3answers
730 views

Why is the algebraic closure of a finite field countable?

An algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. But why is it a countable set?
1
vote
3answers
48 views

About a field of order $2^{n}$ with $n$ an odd integer and an additional property

I'm new in the world of fields (so I don't have any strong theorem at my disposal) and I've got stuck in this problem: Given a field of order $2^{n}$ with $n$ an odd integer and $a,b$ elements ...
1
vote
1answer
55 views

Models of ZFC as Ordered Fields

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ ...
5
votes
1answer
346 views

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

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 ...
1
vote
2answers
68 views

Field extensions and gcd

Let $L|K$ be a field extension and let $u, v \in L$ be algebraic elements over $K$ such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m, n)=1$ then $Irr(v, k)$ is irreducible on $K(u)$. ...
1
vote
0answers
97 views

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
2
votes
1answer
50 views

Does the degree of a field extension depend on the embedding of the base field?

To formulate the question more precisely, let $f$ be a field monomorphism from $F$ to $E$. The extension field $E$ can be considered a vector space over $F$, if scalar multiplication is defined by the ...
2
votes
2answers
39 views

Proof of characterization of splitting fields

I'm trying to prove that if $K$ is a finite field extension of $F$ such that $K$ is the splitting field of some collection $C$ of polynomials in $F[x]$, then every irreducible polynomial in $F[x]$ ...
1
vote
5answers
203 views

Proving a structure is a field?

Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is: Let $F$ be a field and let $G=F\times F$. Define operations of addition ...
2
votes
1answer
55 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that? ...
2
votes
1answer
55 views

understanding roots of polynomials in field extensions

I'm running into a conceptual stumbling block understanding the application of the FHT to field extensions and finding roots, if anyone has any pointers on where I might be misunderstanding. I'm ...
0
votes
1answer
69 views

Looking for a field isomorphic to $\Bbb{Q}$

I am looking for a field that is isomorphic to $\Bbb{Q}$. Could someone kindly give an example, or construct one such field?
4
votes
3answers
80 views

A question about fields containing a copy of $\Bbb{Q}$

When we say a field contains a copy of the field of rational numbers $\Bbb{Q}$, what does this really mean? Does it mean it contains a field isomorphic to $\Bbb{Q}$, or does it mean it contains ...
6
votes
3answers
250 views

Question about fields and quotients of polynomial rings

I don't see how to solve the following problem: Let $R$ be a commutative and unitary ring. If there exists a monic polynomial $f(x) \in R[x]$ so that $R[x]/(f(x))$ is a field, show that $R$ is a ...
3
votes
1answer
29 views

Prove that $K(\alpha)=K(\alpha^6)$ when $[K(\alpha):K]=2011$

Let $L/K$ be a finite extension and let $\alpha \in L$ so that $[K(\alpha):K]=2011$. Prove that $K(\alpha)=K(\alpha^6)$. My idea is as follows: $K \subset K(\alpha^6) \subset K(\alpha)$, therefore ...
1
vote
1answer
93 views

finite transcendence degree and algebraic closure

Let $k$ be an algebraically closed field. Let $K$ be an extension field of $k$ of finite transcendence degree over $k$. Intuitively, it seems to me that $K$ can not be algebraically closed. Is there a ...
3
votes
4answers
94 views

$\mathbb{Q}(\zeta_7)$ subextension of degree $3$

Let $\zeta_7$ be a $7$-th primitive root of unity. Is there a way to determine a subextension of $\mathbb{Q}(\zeta_7)$ that has degree $3$, without making use of Galois theory stuff?
5
votes
2answers
494 views

Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial ...
6
votes
1answer
86 views

Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with ...