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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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? ...
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1answer
49 views

Were there attempts to build a system of numbers where division by a negative number is greater than division by any positive number?

When we decrease the positive denomenator of a ratio (with positive numerator), the value of the ratio increases. The same happens for negative denomenators. But in zero this law is broken. Were there ...
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2answers
36 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 ...
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2answers
50 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.
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1answer
125 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 ...
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1answer
39 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?
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1answer
66 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
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1answer
36 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 ...
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0answers
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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 $\sum a_it^i \to \sum a_i$. ...
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4answers
50 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 ...
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0answers
23 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 ...
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0answers
18 views

Find an example of an infinite body of characteristic p. [duplicate]

Let p is a prime number. Find an example of an infinite field of characteristic p.
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0answers
24 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
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2answers
42 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 ...
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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 ...
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3answers
33 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 ...
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3answers
40 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 ...
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0answers
30 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 ...
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0answers
31 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 ...
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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))= ...
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1answer
27 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 ...
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1answer
98 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 $L$ ...
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1answer
31 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 ...
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2answers
107 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
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1answer
63 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 ...
3
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2answers
91 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 = ...
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3answers
32 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 ...
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1answer
47 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 $\subseteq^{*}$ ...
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168 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 ...
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2answers
52 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 $K(u)$. ...
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0answers
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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 ...
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1answer
28 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 ...
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2answers
21 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 $F[x]$ ...
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5answers
86 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 ...
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1answer
38 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? ...
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1answer
27 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 ...
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1answer
59 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?
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3answers
56 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 ...
4
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3answers
184 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 ...
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1answer
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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 ...
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1answer
31 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 ...
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4answers
76 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?
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1answer
61 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 ...
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1answer
54 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 ...
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1answer
20 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
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1answer
46 views

Multiplicative Property of the degree of field extension

According to Artin's Algebra, chapter 15, section 3, the mapping property of the degree of field extension is as follows: Let $F\subset K\subset L$ be fields. Then $[L:F]=[L:K][K:F]$, where $[K:F]$ ...
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3answers
174 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
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0answers
35 views

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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2answers
34 views

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R-M$ is a unit. Then $R/M$ is a field.

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. I am solving this question of NBHM 2011. To solve this is ...
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1answer
39 views

Separability is transitive … for infinite extensions

Let $L/M/K$ be a tower of fields. The proof that $L/K$ is separable iff $L/M$ and $M/K$ are also separable is contained in a lot of notes and texts I've come across, subject to the assumption that the ...