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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2
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1answer
184 views

Galois group $Gal(\mathbb{Q}[\xi]:\mathbb{Q})$

Lets $\xi$ is primitive n-th root of unity over $\mathbb{C}$ and lets $$\mathbb{Q}[\xi]\ni\alpha=\sum_{i=0}^{n-1}a_i{\xi}^i.$$ Consider any element $g\in Gal(\mathbb{Q}[\xi]:\mathbb{Q})$. $$g(\alpha)=...
5
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2answers
219 views

How many fields inside $\mathbb R$?

i.e. what is cardinality of $\{A \mid \ A\subset \mathbb R, A \text{ is a field} \}$?
4
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1answer
768 views

Linearly disjoint field extensions

Recall that if $k$ is a field, some field extensions $K_1/k$,..., $K_n/k$ are called linearly disjoint if the tensor product $K_1\otimes_k\cdots \otimes_k K_n$ is a field. Let $\zeta_5$ be a pritive ...
5
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1answer
1k views

Frobenius homomorphism

Its easy to proof that any non-zero field homomorphism is injective: Proof Assume that $\exists a, b\in F: a\neq b~~and~~\psi(a)=\psi(b)$ then: $$\psi(1)=\psi((a-b)^{-1}(a-b))=\psi((a-b)^{-1})\cdot 0,...
10
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2answers
2k views

Tensor product and compositum of fields

Let E/k, F/k be two arbitrary field extensions of k. My question is: Is there a field extension M/k s.t. E/k, F/k are subextensions of M/k? Alternatively, can we talk about compositum fields without ...
14
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3answers
5k views

Two finite fields with the same number of elements are isomorphic

Fraleigh(7ed) Theorem33.12. Let $p$ be a prime and let $n\in\mathbb{Z}^+$. If $E$ and $E'$ are fields of order $p^n$, then $E \simeq E'$. Proof in the text: Both $E$ and $E'$ have $\...
27
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4answers
7k views

Do finite algebraically closed fields exist?

Let $K$ be an algebraically closed field ($\operatorname{char}K=p$). Denote $${\mathbb F}_{p^n}=\{x\in K\mid x^{p^n}-x=0\}.$$ It's easy to prove that ${\mathbb F}_{p^n}$ consists of exactly $p^n$ ...
11
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4answers
2k views

How to show that $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}]=9$?

Fraleigh, Sec31, Ex9. Show that $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}]=9$. Here is my trial: It is obvious that $\sqrt[3]2$ is algebraic of degree 3 over $\mathbb{Q}$, since $x^3-2$ is ...
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2answers
478 views

isomorphism between specific generated field and specific quotient ring — gap in a proof

$K'$ is a field extension of $F$, $h\in F[x]$, $h$ is minimal for $u'\in K'$, $F(u')$ is a field generated by $F\cup \{u'\}$, $K'=F(u')$. In [1. XIII. Galois theory. 2. ...
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1answer
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Normal closure of a radical extension is radical

I'm struggling to understand the proof that the normal closure of a radical extension of fields is also a radical extension, which is crucial since it allows us to work with radical and normal ...
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1answer
300 views

A problem about Field extension [closed]

Definition of normal extension: an algebraic extension $K$ of $F$ is normal extension if every irreducible polynomial in $F[x]$ that has one root in $K$ actually splits in $K[x]$. Let $K$ be a normal ...
2
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1answer
105 views

To determine whether a field contains free abelian groups of arbitrarily large finite rank

Suppose that $K$ is an algebraically closed field. There is a statement: If $K$ is not the algebraic closure of a finite field, then $K^*$ contains free abelian groups of arbitrarily large finite ...
3
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1answer
137 views

Do $p^n$-th powers determine the field?

This is a question which came to my mind, when seeing A quick question on transcendence Suppose $F$ is a field of characteristic $p$. Then the set of $p^n$-powers of the elements of $F$ is again a ...
3
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2answers
159 views

A quick question on transcendence

I've seen the following claim in my notes, but I couldn't see why it's true: Suppose that $y \in F_p((x))$ is transcendent over $F_p(x)$, denote $L:=F_p(x, y)$ and let $L^p$ be the field of $p$th ...
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0answers
305 views

the algebraic elements form a field [closed]

I proved this simple thing, but using some simple field theory. I want to know whether I can prove it with simpler tools. The proof is not difficult, it uses only a little field theory, like the idea ...
4
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2answers
2k views

Degree of splitting field of $x^6-3$ over $\mathbb{Q}((-3)^{1/2})$ and also over $\mathbb{F}_5$

I am trying to find the degree of the splitting field of this polynomial over these two fields. For the degree over $\mathbb{Q}((-3)^{1/2})$ I got 3. I am pretty sure this is correct. For $\mathbb{F}...
20
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1answer
2k views

Galois group of a reducible polynomial over $\mathbb {Q}$

Let $f \in\mathbb {Q}[X]$ be reducible - for the sake of simplicity, $ f = gh$ with $g,h \in\mathbb {Q}[X]$ irreducible. Let L be the splitting field of f. Does $Gal(f) \simeq Gal(g) \times Gal(h)$ ...
1
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2answers
261 views

A quick question on polynomial division in fields of characteristic p

Suppose that $L$ is a field of characteristic $p$, $E$ is a field extension of $L$, a is a pth root of an element of $L$ such that a is not in $E$. Consider the polynomial $p(x):=x^p-a^p.$ Question: ...
10
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2answers
652 views

Why is a variety over a non-alg. closed field a hypersurface?

Exercise $3$ on page $8$ of Kunz's Introduction to Commutative Algebra and Algebraic Geometry is as follows: If the field $K$ is not algebraically closed, then any $K$-variety $V \subset A^n(K)$ can ...
10
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3answers
1k views

$F[x]/(x^2)\cong F[x]/(x^2 - 1)$ if and only if F has characteristic 2

Artin's Algebra, Chapter 10 problem 5.16 states: Let $F$ be a field. Prove that the rings $F[x]/(x^2)$ and $F[x]/(x^2-1)$ are isomorphic if and only if $F$ has characteristic 2. As a pedantic ...
5
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2answers
294 views

Is $\mathbb{R}(X+Y)\subseteq\mathbb{R}(X,Y)$ a purely transcendental extension?

Is there a nice, short and elementary argument that the field extension $\mathbb{R}(X+Y)\subseteq\mathbb{R}(X,Y)$ is purely transcendental? Obviously, $\mbox{tr deg}_{\mathbb{R}(X+Y)}\mathbb{R}(X,Y)\...
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2answers
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Find primitive element such that conductor is relatively prime to an ideal (exercise from Neukirch)

This is an exercise from Neukirch, "Algebraic Number Theory", Ch I, Sec 8, Exercise 2, pg 52. It really has me stumped. Suppose $A$ is a Dedekind domain, $K$ its field of fractions, $L$ a finite, ...
6
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1answer
181 views

Special types of extensions of fields

Let $K$ be a field. Let $p$ be any prime number. Can one always construct an algebraic extension $K_p$ of $K$ with the following properties? (1) If $L$ is a finite extension of $K$ contained in $...
4
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3answers
161 views

$F[a] \subseteq F(a)?$

I think this is probably an easy question, but I'd just like to check that I'm looking at it the right way. Let $F$ be a field, and let $f(x) \in F[x]$ have a zero $a$ in some extension field $E$ ...
15
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3answers
620 views

Weird subfields of $\Bbb{R}$

I found this problem, and I can't get an answer to it: Prove that there are subfields of $\Bbb{R}$ that are a) non-measurable. b) of measure zero and continuum cardinality. I can't seem ...
2
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2answers
227 views

Monomorphisms and Isomorphisms

I came across the following problems: If $\varphi: F \to G$ is an isomorphism of fields show that $\varphi^{-1}: G \to F$ is also an isomorphism. So $\varphi(a+b) = \varphi(a) + \varphi(b)$ and $\...
17
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3answers
881 views

Why doesn't stuff hold in characteristic non-zero?

There are a bunch of theorems in algebra that require the underlying field to be characteristic 0. I seem to remember that these all stemmed from one basic fundamental theorem that only holds in ...
4
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0answers
187 views

When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same. Thanks in advance.
3
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1answer
112 views

Set of solutions to quadratics over $\mathbb{Q}$

Does the set of solutions to quadratics over $\mathbb{Q}$ form a subgroup of the additive group $\mathbb{R}$?
13
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2answers
1k views

Grassmann numbers as eigenvalues of nilpotent operators?

The following question is motivated by the construction of the fermionic path/field integral, as done for example in Altland & Simons "Condensed Matter Field Theory". Consider the vector space $\...
87
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3answers
30k views

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
3
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2answers
219 views

Why is a polynomial of form $g(X^{p^m})$ over a field of characteristic $p$ not necessarily inseparable?

A small proposition in Ash's Algebra states that over a field $F$ of prime characteristic $p$, an irreducible polynomial $f$ is inseparable if and only if $f'$ is the zero polynomial, or ...
4
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1answer
249 views

On a characterization of the tamely ramified coverings of the fraction field of a strict Henselian ring

Let $K$ be the field of fractions of a strictly Henselian discrete valuation ring $A$. Following Milne ("Etale Cohomology", Chapter I, Paragraph 5, example e)), $Spec(K)$ is the algebraic analogue of ...
17
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3answers
5k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
26
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3answers
2k views

Intermediate ring between a field and an algebraic extension.

This is an exercise in some textbooks. Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field. The trouble is really ...
2
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1answer
241 views

Prime degree ⇒ linearly disjoint?

If F is a field of characteristic 0 with subfields K, L such that F is the compositum of K and L and [ L : L ∩ K ] is prime, must be K and L be linearly disjoint over L ∩ K? In other words, must [...
3
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2answers
183 views

Fertile fields for roots of unity

Is there a field $K$, an odd prime $p$, and a positive integer $n$, such that $K[ζ] = K[ζ^p]$ where $ζ = ζ_{p^n}$ is a primitive $p^n$th root of unity not contained in $K$? In other words, can a base ...
8
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2answers
383 views

How does extending a field affect matrix similitude? [duplicate]

Suppose we have fields $\mathbb{K}_1, \mathbb{K}_2$ such that $$\mathbb{K}_1 \subset \mathbb{K}_2.$$ Question Let $A, B$ be square $n\times n$ matrices over $\mathbb{K}_1$. If there exists $P_2 \...
1
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1answer
152 views

writing a field as an R module

let $F$ be a field. for which ring $R$, $F$ is an $R$-module. i know already that as an abelian group $F$ is a $\mathbb Z$- module, what else can we say for a general field $F$.
3
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1answer
541 views

The degree of the algebraic closure over the separable closure of an imperfect field

Let $K$ be imperfect, $K^a$ its algebraic closure and $K^{\rm sep}$ its separable closure. Show $[K^a \colon K]$ and $[K^a\colon K^{\rm sep}]$ are infinite. Is $[K^{\rm sep}\colon K]$ infinite? Since ...
2
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2answers
88 views

Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?

I have a question about the some rings and fields. Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
12
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3answers
1k views

What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?

Let $Aut(\mathbb{C}/\mathbb{Q})$ be the field automorphisms of $\mathbb{C}$, and $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$ the subfield of $\mathbb{C}$ fixed by this group. I supsect that it is equal ...
1
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1answer
142 views

Polynomial equations with finite field arithmetic

there are given 3 equations (they are connected with cyclic codes): $$s(x)=v(x)+q(x)g(x)$$ $$g(x)h(x)=x^7+1$$ $$s(x)=v(x)h(x)\bmod(x^7+1)$$ I have following data (for $GF(8)$ with generator ...
4
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1answer
1k views

There are enough Galois extensions?

There are enough Galois extensions? For me enough means that every finite extension of a certain field is included in a Galois extension of that field, formally: "Given a field $k$, and a finite ...
10
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3answers
2k views

Puiseux series over an algebraically closed field

Using the construction $R_n = K[t^\frac1n]$, $L_n = \text{Quot}(R_n)$ and $P = \bigcup_{n\in \mathbb{N}}L_N$ one automatically gets that the Puiseux series are a field. Nevertheless they are also an ...
13
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1answer
396 views

Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?

I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
8
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3answers
2k views

Galois Field Fourier Transform

there are two definitions for Reed-Solomon codes, as transmitting points and as BCH code ( http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction ). On Wikipedia there is written that we ...
0
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2answers
125 views

Binomial formula in $GF(2^m)$

there is a binomial formula: $$(x+y)^n=\displaystyle\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$ When operations are done in $GF(2^m)$ then all positive integers are reduced $\bmod2$, so binomial formula ...
4
votes
3answers
2k views

Primitive polynomials of finite fields

there are two primitive polynomials which I can use to construct $GF(2^3)=GF(8)$: $p_1(x) = x^3+x+1$ $p_2(x) = x^3+x^2+1$ $GF(8)$ created with $p_1(x)$: 0 1 $\alpha$ $\alpha^2$ $\alpha^3 = \...
0
votes
1answer
97 views

Field extension

there is for example field $GF(2^4)=GF(16)$. Is $GF(16)$ a subfield of itself? Following this definition http://mathworld.wolfram.com/Subfield.html there is nothing written that subfield must contain ...