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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Proof that the following polynomial is irreducible

I need to check whether the following polynomial $g(x)=x^2+t^2-1\in\Bbb{C}(t)$ (where $\Bbb{C}(t)$ is the field of all rational expressions over $\Bbb{C}$) is reducible. My attempt: I assume by ...
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Intermediate extension of degree n of the algebraic closure of Q [duplicate]

I need to prove that for every natural number n, there exists an intermediate extension E of A/Q with [E:Q]=n, being A the algebraic closure of Q (the field of all complex numbers that are algebraic ...
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1answer
26 views

Existence of a non-zero T- invariant subspace on $\mathbb{Q}^{4}$

For any linear transformation $T:\mathbb{Q}^{4} \to \mathbb{Q}^{4} $, does there always exist a non-zero T- invariant subspace? As we know For any linear transformation $T:\mathbb{R}^{4} \to \mathbb{...
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1answer
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Zeroes of $x^{p^n} - x$ are distinct in splitting field of $\mathbb{F}_p$

I am attempting to show that the zeroes of $x^{p^n} - x$ are distinct (as a part of showing the splitting field over $\mathbb{F}_p$ has $p^n$ elements). I understand that it is sufficient to show that ...
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When can we embed field extensions into one another?

I was recently informed of the following theorem: Let $K$ be a field of characterisic $0$, and let $L$, $M$ be field extensions of $K$ such that i) $L$ is algebraic over $K$, ii) every irreducible ...
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1answer
36 views

Prove $\text{Hom}(K,K)=\{0,id_K\}$ for $K=\mathbb{Q}$

I've been solving some problems from my Galois Theory course, and I need help with the final step of this one: Prove $\text{Hom}(K,K)=\{0,id_k\}$ for $K=\mathbb{Q}$. The work I did so far: Given $K$ ...
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For which rational coefficients is $a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4$ constructible, where $\alpha=3^{1/5}$?

Let be $\alpha=3^{1/5}$, and be $\gamma$ the number $$ \gamma=a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4 $$ with $a,b,c,d,e \in \mathbb{Q}$ . For which $a,b,c,d,e \in \mathbb{Q}$ is $\gamma$ ...
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Does maximality of $p=A \cap q$ imply maximality of $q$, where $A \subseteq B$ is algebraic?

Let $A \subseteq B$ be two commutative $\mathbb{C}$-algebras. Let $q$ be an arbitrary ideal of $B$ and denote $p:=q \cap A$. By [Lemma 3][1], if $A \subseteq B$ is an integral extension, then: $q$ is ...
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51 views

Showing that $\mathbb{Q}(\sqrt{3+\sqrt{3}})$ is not Galois.

Why $\mathbb{Q}(\sqrt{3+\sqrt{3}})$ is not Galois? I have only this: Let $\alpha=\sqrt{3+\sqrt{3}}.$ If $\alpha=\sqrt{3+\sqrt{3}}$ then $(\alpha^2-3)^2=3$ then $f(x)=x^4-6x^2+6$ is the minimal ...
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An algebraic structure that enables solving radical equations rigorously using that algebraic structure operations only.

My understanding of abstract algebra, and the concept of a field in particular, is that it's an abstract algebraic construction that resembles the real numbers. And that it abstracts out and captures ...
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40 views

Question on Group Actions and one to one correspondence with a field

I am really struggling with this problem: Let $\mathbb{F}$ be a field and $\mathbb{F}^{\times}$ is a group under multiplication that consists of the units in $\mathbb{F}$. We consider the following ...
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2answers
35 views

A cubic polynomial over $\mathbb{Q}$ with a degree three splitting field?

I'm wondering if it's possible to have a cubic polynomial $f(x)\in\mathbb{Q}$ with three distinct roots (in $\mathbb{C}$) that has a degree three splitting field? If $f(x)=(x-a)(x-b)(x-c)$, with $a,b,...
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1answer
27 views

Given examples of complete non-Archimedean field of characteristic $p$ s.t. $K$ is not $F$-finite

Let $K$ be a complete non-Archimedean field of characteristic $p$ with valuation ring $A$. We say a ring $R$ is $F$-finite if it is of characteristic $p$ and the absolute Frobenius map $F_R$ is finite....
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I have read that any integral domain can be extended to be a field. How do I do it?

There is an associative, commutative infinite-dimensional non-Archimedean algebra over $\mathbb{R}$ with no zero-divisors (integral domain), where we cannot divide by some elements. If we try, we get ...
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1answer
70 views

Adjoing $p$th roots to rings: as quotients of polynomials

Motivation/Setup: Suppose $R$ is a domain. I'd want to "adjoin $p$th roots of an element $a \in R$." where $p$ is a prime (which i am mostly interested in $p$-adic) This seems to be a cmomon ...
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2answers
40 views

Group of units in a finite field

Let $f(x) \in \mathbb{Z}/3\mathbb{Z}[x]$ be a cubic irreducible polynomial and let $F = \mathbb{Z}/3\mathbb{Z}[x]/(f(x))$. I want to show that either $x$ or $2x$ generates the group of units of $F$. ...
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37 views

How to define operations $+$and $\cdot$ if a set has $3$ elements?

How to define operations $+ $and $\cdot$ if a set $F$ has $3$ elements {j, k,l } rather than the usual $2$ elements using the table so that it is a field? And more broadly, how can I define operation ...
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30 views

Why does the Nullstellensatz imply that there is a zero of a maximal ideal which is algebraic over a field?

I am reading Ernst Kunz's "Introduction to commutative algebra and algebraic geometry" and I'm not quite understanding why the underlined sentence in the screenshot below is true. The ...
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Prove that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}\sqrt[3]{3})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{3})$

Showing that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})$ has degree 6 over $\mathbb{Q}$ is straighforward: It contains $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{3})$ which are degree 2 and 3 over $\...
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1answer
37 views

What is $[\mathbb{Q}(a) : \mathbb{Q}(a+1/a) ]$?

I've been thinking about this for a while and I keep confusing myself, so hopefully someone can help clarify some thoughts. I'm currently looking at two fields, $\mathbb{Q}(a+1/a)$ and $\mathbb{Q}(a)$,...
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Basic question about inseparable field extension

Suppose $L/K$ is not separable. Does this imply that there is an $a\in L$ such that $a^p\in K$ but $a\not\in K$? I know that the characteristic of the fields must necessarily be $p>0$. By ...
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1answer
51 views

Möbius transformation preserving polynomial

Let $K$ be an algebraically closed field, and $x$ transcendent over $K$. We know that the group of $K$-automorphism of $K(x)$ is $\operatorname{PGL}(2,K)$, whose elements satisfy $ \alpha (x)= \...
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1answer
40 views

Countable algebraically closed field inside an incountable algebraically closed of characteristic zero

I would like to know if the following claim is true. I found this claim in a paper without proof :(. Let $k$ an uncountable algebraically closed field of characteristic $0$. Let $S=\operatorname{Spec}...
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1answer
19 views

K-isomorphic splitting fields in field with characteristic 2

The problem is as follows: Let $K$ be a field of characteristic $2$, let $a$ be an element of $K$ which is not of the form $b^2 + b$ for any $b \in K$, let $f_a(X) = X^2 + X + a$, and let $L_a = K(\...
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19 views

Finding minimal polynomial of an element in an extension over $\mathbb{Q}$

What is the minimal polynomial of $\sqrt2 +\sqrt{2}i$ over the field $\mathbb{Q}$. I proceeded in the following way: Let $x=\sqrt2+\sqrt{2}i \Rightarrow x^2=2+2i^2+4i \Rightarrow x^2=4i \Rightarrow x^...
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intersections of powers of maximal ideal in integral domain

Let $\Omega\subset\mathbb{C}$ be a disc. $p\in \Omega$. In the integral domain $R=H(\Omega)$ " the space of holomorphic functions in $\Omega$, consider the maximal ideal $M_p=\{f \in R : f(p)=0\}$...
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1answer
15 views

When does conjugation in simple field extensions coincide with the identity?

Suppose I have a field $F$ and a simple extension of $F$, $F(\alpha)$, and let's assume that $\alpha^2 \in F$. Now take the "conjugation" automorphism $\sigma:F(\alpha) \to F(\alpha)$, $\...
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26 views

Degree of $\mathbb{Q}(\sqrt{a_1}, \ldots , \sqrt{a_n})$ over $\mathbb{Q}$ for $a_i$ squarefree and $gdc(a_i,a_j)=1$

It seems obvious that $[\mathbb{Q}(\sqrt{a_1}, \ldots , \sqrt{a_n}):\mathbb{Q}]=2^n$,provinding that $a_i$ is squarefree $i=1, \ldots, n$ and $gcd(a_i,a_j)=1, \forall i \neq j $but I am having ...
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22 views

Given f(x) in k[x], is it possible to find two field extensions K/k and K'/k such that f has two different factorizations as linear polynomials?

By Kronecker's theorem, given a field $k$ and $f(x)$ a polynomial with coefficients in $k$, there exists a field $K$ containing $k$ as a subfield and with $f(x)$ a product of linear polynomials in $K[...
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1answer
76 views

Splitting field of a polynomial over Z_5

Find the splitting field of the polynomial $$x^3+x+1$$ over $\mathbb{Z}_5$. I can see that the given polynomial is irreducible over $\mathbb{Z}_5$ So, $$\dfrac{\mathbb{Z}_5 [x]}{(x^3+x+1)}$$ is a ...
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1answer
60 views

What primes factor both $2$ and $3$, with no restriction on the domain? [closed]

What numbers (especially primes) factor both $2$ and $3$ (with no restriction on the domain)? How many answers are there? I'm looking preferably for a prime. My attempt: No natural number (other ...
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0answers
44 views

How would I determine all the cubes in mod 13? [closed]

Just looking how to do this I have been asked to find all cubes in Z_13, which I assume is cubing all elements of Z_13 under mod 13, then what is the remainder is what can be involved in the set?
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40 views

Polynomial of degree $3$ from $F[x]$ has a zero in $F$, then how can I know about its reducibility? [closed]

I have a polynomial of degree greater than $3$ from $F[x]$, F is field has a zero in $F$, then how can I know about its reducibility and irreducibility? e. g, $x^5+x-3$ irreducible or not over $R$? I ...
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1answer
36 views

Suppose $F ⊂ K$ are fields. Let $f(x) ∈ F[x] ⊂ K[x]$. Suppose that $f(x)$ is irreducible in $K[x]$. Prove that $f(x)$ is also irreducible in $F[x]$.

Suppose $F ⊂ K$ are both fields. Let $f(x) ∈ F[x] ⊂ K[x]$. Suppose that $f(x)$ is irreducible in $K[x]$. $a)$ Prove that $f(x)$ is also irreducible in $F[x]$. $b)$ Is it true that if $f(x)$ is ...
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1answer
28 views

Normal open subgroup of $Gal (L / K)$

Let $L / K$ be Galois extension and $H$ be a subgroup of $Gal (L / K)$. Suppose that $K$ is a subfield of $L$ that is invariant by $H$. Show that $HN = Gal (L / K)$ for any normal open subgroup $N$ of ...
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1answer
50 views

Can the minimal polynomial have multiple roots?

Let $F$ be an extension field of $K$. I'm trying to think of an example of an element $\alpha$ which is algebraic over $K$ and whose minimal polynomial over $K$ has a root of multiplicity $> 1$ in ...
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20 views

Understanding why $E(\alpha)$ is a splitting field for $F(\alpha)$ when $E$ is for $F$.

Why if $E$ is a splitting field of a polynomial $f$ over $F$ and $\alpha$ in some extension of $F$ then $E(\alpha)$ is a splitting field of $f$ over $F(\alpha)$? The argument that justifies the above ...
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41 views

Is every zero divisor free ring a field? If that's not the case give an example. [closed]

Is every ring without zero divisors a field? If that's not the case give an example. A zero divisor exist if $x\cdot y =0$, while $x\neq 0 $ and $y\neq 0$. As I'm studying for an upcoming exam I have ...
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32 views

Polynomial reduction modulo p

I'm trying to read this research paper, but there's a section I can't understand. So we have this polynomial: $Q_{k}(z)=z^{6k+7}−8z^{6k+6}+ 26z^{6k+5}−44z^{6k+4}+ 41z^{6k+3}−20z^{6k+2}+ 4z^{6k+1}+ 15z^...
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22 views

Vector spaces and fields, addition and product operators.

Thank you for taking the time to read my question. I am reading about fields and vector spaces. Any vector space $V$ is a vector space over some field $F$. Furthermore, for any set $S$, it is my ...
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3answers
55 views

Field extension of $\Bbb Q$ of degree $4$ has reals

Suppose $L / \mathbb{Q}$ is a field extension with $[L : \mathbb{Q}]$ = 4, with $L \not\subset \mathbb{R}$. Is it true that $L \cap \mathbb{R} \neq \mathbb{Q}$? If not, is it true if $L / \mathbb{Q}$ ...
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2answers
48 views

Let $F$ be a field. Prove that $F$ cannot have two different additive identities [closed]

Let $F$ be a field. Prove that $F$ cannot have two different additive identities (i.e., there cannot be two different elements $a, b \in F$ such that, for every $x \in F$, $a+x = x+a = x = b+x = x = b$...
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1answer
77 views

Why is $F=\{0\}$ a field?

Context: I am new to this. I started my course yesterday. I know that the operations $+$ and $\cdot$ are required to satisfy the field axioms. So how can $F=\{0\}$ be a field? Recall the additive and ...
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41 views

Lie groups over fields of finite characteristic

Does anyone have any good references on Lie groups over fields of finite characteristic? I am trying to find something comprehensive that shows what fails and what succeeds in comparison to Lie theory ...
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1answer
19 views

Fixed field with non-normal subgroup and Galois property

Let $L/K$ be a finite Galois extension with Galois group $G$. Furthermore, let $g \in G$ and $M = L^{\langle g \rangle}$. I know that $\langle g \rangle$ does not always have to be a normal subgroup ...
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1answer
63 views

What are the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$?

I would like to calculate the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$. I know that the elements of $\mathbb{Q}(\sqrt[3]{2})$ have the form of ${a+b\sqrt[3]{2}+c\sqrt[3]{4}}$, where a,b,c $\in \...
2
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0answers
42 views

What is the intuition behind the definiton of a seperable field extension?

so I am currently studying separable extensions in John Fraleigh's abstract algebra book and have come upon this definiton of a separable field extension: A finite extension $E$ of $F$ is a separable ...
3
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1answer
52 views

A property of fields of characteristic $p$ and their extensions

I am currently stuck on the following problem, taken from Herstein's Topics in Algebra, 2nd ed.. It reads: If $F$ is of characteristic $p \neq 0$ and if $K$ is a finite extension of $F$, prove that ...
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1answer
31 views

Relation between $\mathbb{C}$-automorphisms and Möbius transformations.

Today in my Algebraic Ecuations course, the proffesor mentioned that while there is only one field automorphism (the trivial one) in $\mathbb{R}$, there are infinite field automorphisms in $\mathbb{C}$...
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31 views

Which elements of $\mathbb{R}[x]$ have multiplicative inverses ? And is $\mathbb{C}[x]$ a field?

Let $\mathbb{R}[x] = \{a_n x^n + \cdots + a_0 \mid \text{where each $a_i \in \mathbb{R}$ and $n \in \mathbb{N} \cup \{0\}$}\}$. Let $\mathbb{C}[x] = \{a_n x^n + \cdots + a_0 \mid \text{where each $a_i ...

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