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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Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...
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Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
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Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? [duplicate]

Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? (It's well-known that if $F$ is a finite field, $F^*$ is a cyclic group). Thank you in advanced.
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Integral domain (rings and fields)

Let $\mathbb Z[i]=\{a+ib \mid a, b \in \mathbb Z \}$. How to Show that $\mathbb Z[i]$ is a integral domain?
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Prove that $u$ is algebraic over $\mathbb{Q}$ [closed]

Let $u$ be a root of the following equation: $$x^{3}+{\displaystyle \dfrac{1+i}{\sqrt{2}}x^{2}+\dfrac{-1+i\sqrt{3}}{2}x+1=0}$$ Prove that $u$ is algebraic over $\mathbb{Q}$ . Thanks in ...
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1answer
47 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
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1answer
60 views

If $|\operatorname{Aut}_KF|=3$, must we have cube roots of unity?

Let $K$ be a field of zero characteristic. Let $F$ be a finite dimensional extension field of $K$ such that $|\operatorname{Aut}_K F|=3$. Must the equation $x^2+x+1=0$ have a root in $F$ ? Thank you
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34 views

Existence of “simple” irreducible polynomial of degree 12 in a finite field

Assume that we have a finite field $\mathbb{F}_p$, where $p$ is prime, $p \equiv 1\ (\textrm{mod}\ 4)$ and $p \equiv 1\ (\textrm{mod}\ 3)$. I was looking for irreducible polynomial in a form $X^{12} + ...
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38 views

Proving that something is a field

Consider the function $h(x)=x^3+x^2+1$ in $\mathbb{Z}_5[x].$ Prove that $\mathbb{Z}_5[x]/h(x)$ is a field. Is $\mathbb{Z}_5[x]$ algebraically closed?
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characteristic of a field>0

I have no idea how to solve such an exercise: Given is a field K, $ n \in \mathbb{N}, 1_K:=(1,1,..1) \in K^n $ $e_1,..,e_n$ is a standard basis of $K^n$. One show that: $1_K-e_1,..,1_K-e_n$ are ...
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1answer
35 views

Can an algebraic extension of an uncountable field be of uncountable degree?

I want to show that, if $K$ is a maximal subfield of $\mathbb C$ without $\sqrt{2}$ in it, then $\mathbb C$ is of countable degree over $K$. Is it the case that an algebraic extension of an ...
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35 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 ...
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1answer
24 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 = ...
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1answer
69 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 $F$. ...
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20 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 ...
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22 views

Transitivity of norm and trace for intermediate fields of finite separable extensions

Let $K/F$ be a finite separable extension of fields. Show that $N_{K/F} = N_{E/F}\circ N_{K/E}$, and $Tr_{K/F} = Tr_{E/F}\circ Tr_{K/E}$ for any intermediate field K/E/F.
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22 views

Show that $K_1K_2=K_1(K_2)$ is abelian. [duplicate]

Let $L/F$ be a field extension and $L/K_i/F$ with $K_i/F$ abelian. Show that $K_1K_2=K_1(K_2)$ is abelian.
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2answers
29 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 $f = \prod_{i=1}^n(X-\alpha_i)$. ...
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58 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 ...
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1answer
41 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) ...
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3answers
128 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 ...
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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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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
47 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
38 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
65 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 ...
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1answer
34 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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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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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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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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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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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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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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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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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
39 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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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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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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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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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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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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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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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 $F$ ...
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61 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 ...
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81 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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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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143 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 ...