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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2
votes
1answer
178 views

What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
0
votes
2answers
59 views

Is $t^{2}$ a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$?

I wish to find out if $t^{2}$ is a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$ so I can justify the use of Eisenstein on the polynomial $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$ I believe ...
0
votes
2answers
64 views

Irreducibility of $x^{3}-t\in\mathbb{C}(t)[x]$

Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$ Is it true that $p$ is irreducible over $F$ ? My thoughts: I think that since it is not true that $t^{2}\mid t$ (I don't know how to ...
0
votes
0answers
59 views

Is this a legitimate definition of a automorphism of a simple extension $F(\zeta)$?

I am trying to understand the objection my lecture made to my definition, this is the case: We have it that $F$ is a field s.t. $char(F)\neq5$ and $\zeta$ is a root of $$x^{4}+x^{3}+x^{2}+x+1\in ...
1
vote
2answers
179 views

Irreducible polynomial substitution

Let $F$ be a field. Suppose that the polynomial $p(x,y)$ is irreducible in $F[x,y]$. Let $a(x)\in F[x]$ be a polynomial of positive degree. Prove that $p(a(x),y)$ is irreducible in $F(a(x))[y]$. I ...
5
votes
1answer
252 views

An angle $\theta$ can be trisected if and only if $4x^3-3x+\cos\theta$ is reducible over $\mathbb{Q}(\cos\theta)$

I was given the above problem for homework. There is (what seems to be) a relevant proof in my textbook regarding the impossibility of trisecting $\pi/3$. In this proof, the identity $$\cos 3\theta = ...
1
vote
2answers
312 views

Proving that a set is not an ordered field

Problem: Let $S=\left \{ 0,1,2 \right \}$. How can someone prove that there is unique way of defining addition and multiplication such that $S$ is a field if $0$ of the set $S$ has the meaning (for ...
3
votes
1answer
114 views

Linear Transformation over Subfield

Letting $F\subseteq K$ be fields, and $V$ a vector space over $K$. Certainly, $V$ is also a vector space over $F$. And if $\{e_1,...,e_n\}$ is a basis for $K$ over $F$ and ...
0
votes
1answer
81 views

Real embedding of the splitting field of $X^3-2$

Does the splitting field of $X^3-2$ have a real embedding?
1
vote
1answer
280 views

Finitely Generated Algebra and Finite Extension

Suppose $L,K$ are fields. Is is true that if $L$ a finitely generated $K$-algebra then $L/K$ is a finite field extension? Wikipedia seems to think so. But if it is true surely it's difficult to ...
0
votes
1answer
54 views

What is known about moduls $M = F^n$ over a ring $R$ where $F = R/I$ is a field

If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$. If you let $M = F^n$, and define $(\cdot): (R,M) ...
5
votes
3answers
495 views

fields are characterized by the property of having exactly 2 ideals [duplicate]

Possible Duplicate: A ring is a field iff the only ideals are $(0)$ and $(1)$ Michael Artin's Algebra in the introduction of maximal fields, there was a sentence stated that fields are ...
1
vote
1answer
221 views

Non-trivial finite purely inseparable extension

QUESTION: Suppose K/F is a non-trivial finite degree purely inseparable extension. Prove that there is a purely inseparable degree p extension of K. I know, or can prove, that [K:F] is a power of p ...
3
votes
2answers
433 views

Alternative proof that set of algebraic elements is a subfield

Let $E/F$ be a field extension. Let $A$ be the set of all elements in $E$ that are algebraic over $F$. One can show that $A$ is a subfield of $E$. The proofs I read all argue as follows: Let $a,b \in ...
0
votes
0answers
92 views

$x^n-x$ and some irreducible factors properties over $K[x]$

Let $K$ be a field , $a\in K$ , let $d$ be the greatest common divisor, of all the irreducible factors of $x^n-a$ in $K[x]$. $i)$ Prove that $ d|n$ , and there exist $b\in K$ , such that $a^d =b^n$. ...
3
votes
1answer
35 views

Chracterizing quartic polynomials F such that $F, F',F''$ have only real rational roots.

When designing friendly problems for a calculus class one comes up with such a question. (The cubic case is relatively easy.) Of course one can generalize: Characterize degree $n$ polynomials such ...
1
vote
1answer
154 views

minimal polynomial of $x$ over $ K\left(\frac{p(x)}{q(x)}\right) \subset K(x) $

Let $K$ be a field , let's consider the field of rationals functions over x , $k(x)$. Let $t\in k(x)$ be the rational function $\frac{p(x)}{q(x)}$ , where $P,Q$ have no common factors. I have to prove ...
0
votes
1answer
49 views

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?.

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?. I don't know why is necessary a condition for $q$.
1
vote
1answer
399 views

Is the field extension $K(t):K$ normal? Is separable?

The field extension $K(t):K$ where $K$ is any field. I don't know how to apply definition of normal and separable in a transcendental case.
2
votes
1answer
193 views

searching the fixed field of an automorphism, and a primitive generator , in characteristic p.

Let $K$ be a field oh characteristic $p$. Let's take $\sigma \in \operatorname{Aut}(K(x),K)$ where $x$ is trascendental over $K$, where $\sigma(x)=x+1$. Find a primitive element of the fixed field of ...
3
votes
1answer
158 views

separable closure , the separable elements are closed under operations

Let $ K \subset L $ be a field extension. Consider the separable closure in $L$ $$ K_s = \left\{ {x \in L|x\,\,algebraic\,and\,separable\,over\,K} \right\} $$ Prove that $K_s$ is a field. I know ...
10
votes
2answers
247 views

Can all polynomials of a given degree be reducible?

Let $n > 1$ be a fixed integer. Does there exist a field $F$ with the following properties? $F$ is not algebraically closed. Every polynomial $f(x) \in F[X]$ of degree $n$ is reducible. I ...
-1
votes
3answers
188 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
2
votes
2answers
97 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
1
vote
2answers
330 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
3
votes
1answer
227 views

Cyclic Algebras over local fields- reference request

I am looking for an introductory text to the subject of Cyclic Algebras, and in particular ones defined over a local field. A cyclic Algebra, to the best of by understanding is defined as follows: ...
2
votes
5answers
407 views

Example of a union of subfields that is not a field

I'm looking for a field $F$ such that for subfields $K,E$ of $F$ the structure $K\cup E$ is not a field. Clearly both $K$ and $E$ must be proper subfields, and while looking for the answer I've come ...
6
votes
1answer
170 views

Finiteness of the Algebraic Closure

Let $\mathbb R$ be the field of real numbers. Its algebraic closure, the field $\mathbb C$, is a finite extension of $\mathbb R$, which has degree 2. Are there other examples of fields (not ...
4
votes
2answers
471 views

The endomorphism of field

Can we find a field $K$ and an endomorphism $f$ of $K$, such that $f$ is not trivial and $f$ is not surjective? In other words, can we find an endomorphism of $K$ which is not an automorphism?
1
vote
0answers
124 views

splitting field extending an automorphism

Let $\varphi:K_1\to K_2$ be an isomorphism of fields. Let's consider a polynomial $p(x)\in K_1[x]$, and it's splitting field extension $ E_1/K_1 $, also the polynomial $\varphi(p(x))$ $\in K_2[x]$, ...
5
votes
1answer
90 views

On the existence of a “universal” field without algebraic elements

Let $\mathfrak{M}$ be an infinite cardinal. Consider all fields $F$ which have the following properties: (1) $F$ contains $\mathbb{Q}$. (2) $F$ has cardinality $\leqslant \mathfrak{M}$. (3) All ...
3
votes
1answer
286 views

“Place” vs. “Prime” in a number field.

I have been trying to make sense of what a "place" is. In the setting of a number field, is a place simply a prime ideal? My understanding is that one can complete a number field with respect to a ...
2
votes
1answer
178 views

Field Extension of the Rationals

So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things: ...
3
votes
3answers
165 views

Exercise on simple extensions

Let $E/K$ be an extension and $L_1,L_2$ intermediate fields of $E/K$ with $L_i:K$ finite. Then necessarily $L_1L_2:L_2$ $\le$ $L_1:K$. Prove $L_1L_2:L_2$ is not necessarily a factor of $L_1:K$. Hint: ...
2
votes
1answer
180 views

What is wrong with this proof of Wedderburn's little theorem?

Wedderburn's little theorem $\quad$ every finite domain $A$ is a field. Proof $\quad$ Let $x$ be a nonzero element of $A$. Because $A$ is finite, there exist positive integers $n$, $k$ such that ...
7
votes
1answer
185 views

the number field $\mathbb{Q}(\cos \frac \pi n)$

Let $x$ be $\cos \displaystyle \frac \pi n$ for some natural number $n$. Then is it true that $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?
2
votes
1answer
274 views

Let $p$ and $q$ be two distinct primes. Pick the correct statements from

Let $p$ and $q$ be two distinct primes. Pick the correct statements from the following: a. $Q(\sqrt p)$ is isomorphic to $Q(\sqrt q)$ as fields. b. $Q(\sqrt p)$ is isomorphic to $Q(\sqrt{−q})$ as ...
2
votes
0answers
289 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
2
votes
1answer
161 views

Linearly disjoint vs. free field extensions

Consider two field extensions $K$ and $L$ of a common subfield $k$ and suppose $K$ and $L$ are both subfields of a field $\Omega$, algebraically closed. Lang defines $K$ and $L$ to be 'linearly ...
7
votes
1answer
155 views

Exercise on finite intermediate extensions

Let $E/K$ be a field extension, and let $L_1$ and $L_2$ be intermediate fields of finite degree over $K$. Prove that $[L_1L_2:K] = [L_1 : K][L_2 : K]$ implies $L_1\cap L_2 = K$. My thinking ...
2
votes
0answers
38 views

Real embeddings and linear disjointness

Suppose I have two Galois extensions $F_1, F_2$ of $\mathbb{Q}$ such that $F_1$ has a real embedding. Then is there a general condition on $F_2$ such that $F_1$ and $F_2$ will be linearly disjoint ...
19
votes
1answer
387 views

When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?

Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra: ...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
5
votes
2answers
182 views

Does $\mathbb{Z}_5(\sqrt[3]{3})$ make sense? Or, can we always extend a field by a root of a reducible polynomial?

I'm preparing assignment questions for a course in ring/field theory. We'll shortly be looking at extension fields, and the students are meant to understand what notation such as $\mathbb{R}(i)$ ...
0
votes
1answer
89 views

Contracting an angle (using straightedge and compass)

In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable. Shouldn't this be $\frac{\pi}{n}$ instead of ...
5
votes
1answer
469 views

Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$

Following my previous question the book then asks Use this to determine when the field extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$ is biquadratic (where $a,b\in\mathbb{Q}$) ...
12
votes
4answers
737 views

Proving $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}\iff a^{2}-b$ is a square

This is an exercise for the book Abstract Algebra by Dummit and Foote (pg. 530): Let $F$ be a field of characteristic $\neq2$ . Let $a,b\in F$ with $b$ not a square in $F$. Prove ...
3
votes
5answers
464 views

Finding the degree of $1+\sqrt[3]{2}+\sqrt[3]{4}$ over $\mathbb{Q}$

This is an exercise for the book Abstract Algebra by Dummit and Foote (pg. 530): Find the degree of $\alpha:=1+\sqrt[3]{2}+\sqrt[3]{4}$ over $\mathbb{Q}$ My efforts: I first try to find the minimal ...
5
votes
3answers
301 views

Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible

This is an exercise for the book Abstract Algebra by Dummit and Foote (pg. 519): show $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible for $a\neq-1,0,2$ I need help with this exercise, I don't have an ...
3
votes
1answer
341 views

Computing the inverse of an element in $\mathbb{Q}(\theta)$

The polynomial $p(x):=x^{3}-2x-2\in\mathbb{Q}[x]$ is irreducible hence $\mathbb{K=Q}[x]/\langle p(x)\rangle$ is a field, let $\theta\in\mathbb{K}$ be a root of $p(x)$. Can someone please help me ...
3
votes
1answer
863 views

Proving $1 > 0$ using only the field axioms and order axioms

How do I prove $1 > 0$ using only field axioms and order axioms? I have tried using the cancellation law, with the identities in a field and I cannot get anywhere. Does anybody have any ...