# 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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### 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 ...
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### 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 $Gal(f) \simeq Gal(g) \times Gal(h)$ ...
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### 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: ...
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### 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 ...
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### $F[x]/(x^2)\cong F[x]/(x^2 - 1)$ 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 ...
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### $F[a] \subseteq F(a)?$

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 $E$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 [...
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### 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 ...
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 $P_2 \... 1answer 152 views ### writing a field as an R module let$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$. 1answer 541 views ### The degree of the algebraic closure over the separable closure of an imperfect field Let$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]$infinite? Since ... 2answers 88 views ### Is$\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}]]$? 3answers 1k views ### What is$\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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 3answers 2k views ### Primitive polynomials of finite fields there are two primitive polynomials which I can use to construct$GF(2^3)=GF(8)$:$p_1(x) = x^3+x+1p_2(x) = x^3+x^2+1GF(8)$created with$p_1(x)$: 0 1$\alpha\alpha^2\alpha^3 = \...
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 ...