Questions tagged [field-theory]

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 topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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10 views

Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?

We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. ...
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23 views

Integral closure of a field with uneven characteristic

I am studying a lof of algebra lately and i am stuck on this problem. Let $K$ be a field with uneven characteristic and $E = K(X)(\alpha) $ where $\alpha^2= X^2 -1 \in K(X)$ I am struggling to find ...
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Find splitting field of $(x^3-x^2-x)(x^4-x^2+1)$ over $\mathbb{F}_3$

As written in the title, I have to compute the splitting field of $$(x^3-x^2-x)(x^4-x^2+1)$$ over $\mathbb{F}_3$ I'd like to understand if my attempt is correst, or if I'm missing something. Here's ...
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2answers
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Integer basis for $\mathbb{Z}_{\mathbb Q(\alpha)}$

So we are studying integral field and i am stuck on this problem. Let $f(X) = X^3 -X -12 \in \mathbb{Q}[X]$ and $\alpha \in \bar{\mathbb{Q}}$ be a zero of $f(X)$. I already showed that $f(X)$ is ...
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1answer
30 views

Find irreducible factors without factorizing [closed]

I have an exercise from my course notes that states: Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it) I see immediately that the $...
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1answer
11 views

Confusion Over Polynomial Separability Example

In lecture notes I am reading for Abstract Algebra, in the Separability section for Field Theory, it gives the introduction and definition that a separable polynomial is one that has no repeated roots ...
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0answers
22 views

Convergent power series and Euclidean domain, or field.

Consider the following power series (as a function of $z$): $$G(z) = \sum_{i=0}^\infty a_i z^{-i},$$ where $a_i$ is a real coefficient and $z$ is any complex number. It also satisfies that $$\sum_{...
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1answer
23 views

Number of maximal ideal in the quotient ring [closed]

How to find the number of maximal ideal in the Quotient ring $\frac {\mathbb{Z}_5{[x]}}{<(x+1)^2(x+2)^3>}$. ? $R/A$ is an integral domain iff $A$ is prime.
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1answer
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Is there an isomorphism of fields between $\mathbb{F}_{3^{2}}$ and $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$?

if $\mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}$ where $i=\sqrt{2}=\sqrt{-1}$ and we define $(a+bi)+(c+di):=(a+c)+(b+d)i$ and $(a+bi)\ast (c+di):=(ac-bd)+(ad+bc)i$ Is there an isomorphism of fields ...
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1answer
27 views

Unique subfield $L$ of $K=\mathbb{Q}(\sqrt[5]{2},\zeta_5)$ such that $[K:L]=5$

$K$ is the splitting field of the polynomial $f=x^5-2$, and we need to prove the existence of a unique subfield $L$ such that $[K:L]=5$. It would be nice to use the Galois group here, but it is a semi-...
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1answer
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Definition of non split Cartan subgroup

What is the definition of the non split Cartan subgroups of $GL_2(\mathbb{F}_p)$? And what are the explicit expression of a matrix of this subgroups? I read on "Modular Functions of One Variable III" ...
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3answers
57 views

Proving F is a field [closed]

Let $F=\{a+bi; a,b \in F_3\}$ where $i=\sqrt{2}=\sqrt{-1}$ and we define $(a+bi) + (c+di) := (a+c)+(b+d)i$ and $(a+bi) * (c+di):= (ac-bd)+(ad+bc)i$ with $0=0+i$ and $1=1+0i$ Prove $F$ is a field
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0answers
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Show this linear map over a finite field is not the zero map [duplicate]

Let $q=p^k$ be a prime power, let $m\in\mathbb{N}$ and consider the map $$ \tau:\mathbb{F}_{q^m}\longrightarrow\mathbb{F}_{q^m},\quad a\longmapsto\sum_{i=0}^{m-1} a^{q^i}. $$ Show that $\tau$ is ...
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4answers
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For which $n$ is $\sum_{i=0}^n x^i\in\mathbb{Q}[x]$ irreducible?

I know that when $n+1=p$ is prime, $f=\sum_{i=0}^{p-1} x^i$ is the minimal polynomial of $\zeta_p$, a primitive $p$-th root of unity, hence is irreducible. This can be shown by applying Eisenstein to $...
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4answers
357 views

Understanding Fraleigh's proof of: Every finite integral domain is a field

Here's how Fraleigh proves: Every finite integral domain is a field in his book: Let \begin{equation*} 0, 1, a_1, \dots, a_n \end{equation*} be all the elements of the ...
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1answer
22 views

Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$

Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ ...
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1answer
32 views

Degree of extension $\mathbb{C}/K$, where $K$ is maximal with the property $\sqrt{2} \notin K$

This question has been asked before but not really answered, but my query is a bit separate. To summarise the details: $K$ is a field maximal with respect to the property $\sqrt{2}\notin K$, any ...
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2answers
104 views

If the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?

I have a question: if the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?
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0answers
28 views

Degree of an exclusionary Field Extension

Let's say I've got a field $\mathbb{Q}[i]$\ $\mathbb{Q}$. What's the degree of the field extension $\mathbb{Q}[i]$\ $\mathbb{Q}$ : $\mathbb{Q}$? Clearly without the exclusion this has a degree of 2; ...
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1answer
32 views

$f,g \in k[t]$ with $k(f,g)=k(t)$, $\deg(f)=2$ and $\deg(g)=3$

Let $f=f(t),g=g(t) \in k[t]$, $k$ is a field of characteristic zero. Assume that the following two conditions are satisfied: (i) $\deg(f)=2$ and $\deg(g)=3$. (ii) $k(f,g)=k(t)$. Question: Is it ...
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0answers
31 views

Why a polynomial is irreducible over Z

Prove that the following polynomial is irreducible over $\mathbb{Z}$: $$f(x) = x^8-x^7+x^5-x^4+x^3-x+1$$ My attempt: one can see that $f(x)=(x^4+x^3+1)(x^4+x+1) $ over $\mathbb{F_2}$ where these 2 ...
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1answer
32 views

Galois group of field extension

I was asked to find the Galois group of the extension $\mathbb{Q}(\sqrt[3]{2},\sqrt{2},e^{\frac{2\pi i}{3}})$. Since the degree of the minimal polynomials of $\sqrt[3]{2},\sqrt{2}$ and $e^{\frac{2\pi ...
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1answer
49 views

Finding a minimal polynomial of a root of unity over a field extension

I'm trying to find the minimal polynomial of the seventh root of unity over the field $Q(i\sqrt{7})$. I know how to do this over the rationals and have proceeded to finding that $(x-1)(x^6 +x^5 +x^4 +...
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1answer
73 views

$f,g \in k[t]$ such that $\deg(f)=\deg(g)$ and $k(f,gt)=k(t)$

Let $f=f(t), g=g(t) \in k[t]$ be two nonzero polynomials over a field $k$ of characteristic zero. Assume that the following two conditions are satisfied: (i) $\deg(f)=\deg(g) \geq 1$. (ii) $k(f,gt)=...
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1answer
28 views

How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$

This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$. After reading this, I am still confused on how to find ...
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1answer
33 views

Write de addition and multiplication tables for this field [closed]

F3 is the set of residue class of 3 how can i do the multiply and addition table for this field of 9 elements
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26 views

Does adjoining an element to a field change the characteristic? (No it doesn't.)

This may have an answer somewhere already, but I can't find it. Let $K$ be a field of characteristic $p$. Let's adjoin, for example, $\zeta_n$ where $\zeta_n$ is some primitive $n$-th root of unity. ...
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2answers
34 views

(Proof verification) there is no homomorphism between a finite field’s additive group to its multiplicative group.

Given a finite field F with additive group $\text{F}^+$ and multiplicative group $\text{F}^{\times}$ Show that there doesn’t exist $f$:$\text{F}^+ \to \text{F}^{\times}$ s.t. $f(x+y)=f(x)f(y)$. Proof ...
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1answer
41 views

If $K(\zeta,\beta)/K$ is the splitting field for $f$, what can we say about $K(\zeta,\beta)/K(\zeta)$?

Problem setup: Let $n\in\mathbb{N}$ and let $K$ be a field whose characteristic does not divide $n$. The splitting field of $f=x^n-c$ ($c\not=0$) over $K$ is $K(\zeta,\beta)$ where $\zeta$ is a ...
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1answer
25 views

Equivalence between integral closures

I am struggling with the following problem. Let $S$ be a Integral domain and $K$ its Quotient field. Furthermore let $A/B/K$ be Field extensions. We set $X = \text{Int}_S(B)$ where $\text{Int}_S(B)$ ...
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0answers
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Question regarding p-basis in complete local ring

Let $k$ be the residue field of characteristic $p>0$ of a complete local ring $R$ with maximal ideal $m$. A $p$-basis for $k$ is a subset $B \subset k$ such that (i) $k^p(B)=k$ (ii) all ...
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1answer
33 views

Finding class number of quadratic number field using Minkowski bound

My understanding of this is as follows: In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$. Minkowski ...
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1answer
29 views

Real or imaginary part of a field extension of $\mathbb{Q}$.

When $L/\mathbb{Q}$ is a finite extension of $\mathbb{Q}$ that doesn't lie in $\mathbb{R}$. Is there anything we can say about the degree of the extension $L/ L \cap \mathbb{R}$? I know that for ...
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1answer
30 views

$a$ is a norm in $K(\sqrt{a})/K$ if and only if it is the sum of two squares

$a$ is a norm in a quadratic extension $K(\sqrt{a})/K$ if and only if $a=u^2+v^2$ for some $u,v\in K$. $K$ an arbitrary field. Only if follows from: If $a$ is a sum of squares then there exists an ...
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0answers
37 views

Find basis of $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$ over the field $\mathbb Q(\sqrt[3]{2}\omega)$.

Splitting field for the polynomial $\ x^3-2$ over $\mathbb Q$ is $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$. Now roots of the above polynomial are $\sqrt[3]{2}, \sqrt[3]{2}\omega, \sqrt[3]{2}{\omega}^2$ Since ...
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1answer
31 views

Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

Let $K$ be a field of characteristic $p > 0$, and $f ∈ K[X]$ monic and irreducible with root $\alpha$. Let $F$ be the Frobenius endomorphism. To demonstrate: $\alpha ∈ F[K(\alpha)] \Rightarrow f ∈...
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3answers
67 views

Galois group of $x^p-a$ over $\mathbb{Q}$

I have found that the Galois group $G$ of $f=x^p-a$ over $\mathbb{Q}$ is of order $p(p-1)$. I need to show that if $P$ is a subgroup of $G$ of order $p$, then $P$ is normal and $G/P$ is cyclic. ...
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2answers
955 views

Which field property enables us to multiply on both sides by the same value, while preserving equality? [duplicate]

I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off. I was ...
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1answer
41 views

Calculate order of multiplicative group of finite field

How can one calculate the order of a multiplicative group of a finite field such as: $(\mathbb{F}(2^3) \backslash \{0\}, \times)$ Is it as simple as doing $2^3-1$ ?
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1answer
24 views

Is every field a field extension of some form.

I am new to finite field theory .While I was going through the theory I figured that $\mathbb{C}$ is in fact $\mathbb{R}(i) $ isomorphic to $\mathbb{R}[x]/(x^2+1)$ .So I had a question in mind is ...
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0answers
42 views

Can Field of fractions be Algebraically Closed?

Let R be an integral domain which isn't a field. Can it be the case that the field of fractions is algebraically closed? The reason I'm asking this is : The field of fractions of $\mathbb Z$ is $\...
3
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3answers
76 views

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

Let $\mathbb{F}_q$ denote a field with $q=p^n$ elements, where $p$ is prime. Consider the polynomial $f=x^{p^{n-1}}+\ldots+x^p+x$ and the sets $$ \begin{align*} S&=\{a^p-a:a\in\mathbb{F}_q\},\\ ...
3
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0answers
99 views

Galois Group of $x^{6}-2x^{3}-1$

I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt[3]{1+\sqrt{2}}$. I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
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1answer
20 views

Express the fixed field of a subgroup $H$ of $\Gamma(L:K)$ in terms of a basis of $L$ over $K$.

I've been trying to solve this problem for $L:K$ finite with char $K = 0$: $\beta_1,...,\beta_n$ basis for $L$ over $K$, $H$ subgroup of $\Gamma(L:K)$ $\implies$ $\phi(H) = K(\gamma_1,...,\gamma_n)$, ...
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1answer
28 views

Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

$f(x)=2x^3+ax^2+bx+c$ where $a, b, c∈ℤ$. Prove that f is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $d∈ℤ$. Using a hint from my professor I have attempted a proof but I don't think ...
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1answer
38 views

$X^{p^k} - a ∈ K[X]$ irreducible?

Let $K$ be a field of characteristic char$(K) = p > 0$, and let $a ∈ K$ be an element with the following property: $$(\forall \beta ∈ K)(\beta^p ≠ a). $$ Let $k ∈ ℕ$ be arbitrarily given. Is it ...
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1answer
39 views

How to show $E\otimes_k\bar k$ has at least two prime ideals? [closed]

Suppose $k\subsetneqq E$ are two fields, and $E$ is separable over $k$, $\bar k$ is the algebraic closure of $k$, how to show $E\otimes_k\bar k$ has at least two prime ideals?
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1answer
48 views

Product of the elements in Galois group and irreducible polynomial

Statement) Let $K$ be a Galois extension over a field $F \ \text{s.t.} \ \vert G(K/F) \vert =[K;F] =n $ Say $ G(K/F) = \{ \sigma_1(=id) , \sigma_2,..., \sigma_n \}$ Take a element of $\alpha \in K ...
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1answer
36 views

Use of irreducible polynomial in finite field construction

When constructing a finite field $\mathrm{GF}(p^n)$ using polynomials: Why do we need to modulo an irreducible polynomial? What happens if this polynomial is reducible? Does such an irreducible ...
3
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1answer
62 views

Find an irreducible polynomial in $ Q[x]$ of degree $726$.

Find an irreducible polynomial in $ Q[x]$ of degree $726$. I first thought of $x^{726}+1$ to start but its roots would be complex so not in $Q$. Now I'm thinking to use Eisenstein, so taking the ...

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