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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5
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2answers
201 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
votes
1answer
435 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
votes
1answer
622 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 ...
7
votes
2answers
1k 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 ...
8
votes
2answers
2k 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 ...
10
votes
4answers
1k 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 ...
1
vote
2answers
285 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. ...
5
votes
1answer
457 views

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 ...
-1
votes
1answer
233 views

A problem about Field extension

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
votes
1answer
93 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
votes
1answer
130 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
votes
2answers
154 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 ...
2
votes
0answers
197 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
votes
2answers
784 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 ...
12
votes
1answer
699 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
vote
2answers
195 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: ...
9
votes
2answers
420 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 ...
8
votes
3answers
674 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
votes
2answers
214 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 ...
16
votes
2answers
566 views

Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
5
votes
2answers
805 views

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
votes
1answer
145 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
votes
3answers
150 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 $E$ ...
2
votes
2answers
195 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 ...
11
votes
3answers
521 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 ...
3
votes
0answers
141 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
votes
1answer
107 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 $\mathbb{R}$?
12
votes
2answers
850 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 ...
3
votes
2answers
134 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
votes
1answer
167 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 ...
11
votes
3answers
1k 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 ...
10
votes
2answers
376 views

Necessarily a field between a field and its 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 with ...
2
votes
1answer
158 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, ...
2
votes
2answers
132 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 ...
6
votes
2answers
218 views

How does extending a field affect matrix similitude?

Suppose we have fields $\mathbb{K}_1, \mathbb{K}_2$ s.t. $\mathbb{K}_1 \subset \mathbb{K}_2$. On a paper I'm reading there is a flashy reference to some algebraic results concerning similitude of ...
1
vote
1answer
128 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 $F$.
2
votes
1answer
294 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? ...
1
vote
2answers
81 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 $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
4
votes
2answers
459 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
vote
1answer
129 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
votes
1answer
367 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 ...
5
votes
3answers
808 views

Algebraic Closure of Puiseux Series

Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ 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
votes
1answer
343 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 ...
4
votes
2answers
885 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
votes
2answers
92 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 ...
3
votes
3answers
795 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
84 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 ...
1
vote
0answers
172 views

Prime ideal splitting in field extension and its normal closure

The question is: Let L / K be a finite (not necessarily Galois) extension of algebraic number fields and N / K the normal closure of L / K. Show that a prime ideal p of K is totally split in L if and ...
3
votes
0answers
689 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
-1
votes
1answer
190 views

Finite extension of $\mathbb Q_p$

Let $\mathbb K/\mathbb Q_p$ be a finite extension of $p-$adic field $\mathbb Q_p$. Let ${\mathcal O}=\{x\in K\;:\;|x|\leq1\}$ and ${\mathcal P}=\{x\in K:\;|x|<1\}$, here $|\cdot|$ is the absolute ...