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 ...

learn more… | top users | synonyms

7
votes
1answer
242 views

Problem 18.7 in I. Martin Isaacs' Algebra

I am trying to solve the following problem in I. Martin Isaacs' Algebra: A graduate course, p.290: Let $f(X),g(X) \in F[X]$ and suppose $E \supseteq F$ is the splitting field both for $f(X)$ and ...
7
votes
1answer
167 views

If $K=K^2$ then every automorphism of $\mbox{Aut}_K V$, where $\dim V< \infty$, is the square of some endomorphism.

I have to show the following: Let $K$ be a field such that $\mbox{char } K \neq 2$ and each element of $K$ is a square (i.e. $K^2=K$) and let $V$ be a finite-dimensional vector spaces over $K$. ...
7
votes
0answers
122 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
7
votes
0answers
78 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
6
votes
4answers
521 views

Does there exist a field which has infinitely many subfields?

Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields? I don't know how to begin.
6
votes
3answers
447 views

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$?

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$? I think so because $$\mathbb{Q}[\sqrt{2}+1] = \{\sum_{i=0}^{n}c_i(\sqrt{2}+1)^i\mid n\in\mathbb{N}, c_i\in\mathbb{Q}\}$$ $$= ...
6
votes
4answers
469 views

Why is the product of all units of a finite field equal to $-1$?

Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by ...
6
votes
2answers
176 views

Spec of tensor product of fields

Suppose $K/k$ is a finite separable extension of degree $n$. How to show that there exists a finite separable extension $k'/k$ such that $\operatorname{Spec}(K \otimes_k k') $ consists of $n$ ...
6
votes
3answers
221 views

What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
6
votes
5answers
241 views

Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$?

I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?
6
votes
3answers
273 views

Is there a less ad hoc way to find the degree of this extension?

I want to find the degree of $\mathbb{Q}(\sqrt{3+2\sqrt{2}})$ over $\mathbb{Q}$. I observe that $3+2\sqrt{2}=2+2\sqrt{2}+1=(\sqrt{2}+1)^2$ so $$ ...
6
votes
3answers
718 views

How to show that a finite comutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
6
votes
1answer
327 views

Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?

Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ? My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism... What is the 'standard' ...
6
votes
5answers
144 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
6
votes
4answers
372 views

First examples in Galois theory

I'm studying Field Theory and after studying theorems and problems about extensions, splitting fields, etc... I'm starting with the first theorems of the Galois Theory itself. In order to see if I ...
6
votes
3answers
834 views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
6
votes
2answers
344 views

Why aren't there any coproducts in the category of $\bf{Fields}$?

Well the question is stated in the title. I dont know much about field theory and i was suprised when i read it on wikipedia please provide some examples thanks in advance
6
votes
2answers
170 views

In an ordered field, must 1 be positive?

In an ordered field, must the multiplicative identity be positive? Or must it be defined as such?
6
votes
3answers
911 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 ...
6
votes
4answers
90 views

Product polynomial in $\mathbb{F}_7$

I need to compute the product polynomial $$(x^3+3x^2+3x+1)(x^4+4x^3+6x^2+4x+1)$$ when the coefficients are regarded as elements of the field $\mathbb{F}_7$. I just want someone to explain to me what ...
6
votes
3answers
3k views

what is difference between a ring and a field

The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ...
6
votes
5answers
270 views

why are subextensions of Galois extensions also Galois?

An algebraic extension of fields $L|K$ is defined to be a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$. Apparently in the sequence of field ...
6
votes
2answers
997 views

How to prove that $\cos (2\pi/n)$ is algebraic?

Prove that for all integers $n$, $\cos (2\pi/n)$ is an algebraic number.
6
votes
3answers
677 views

On a formula of the norm of an element of a finite extension of a field

Theorem Let $F$ be a field. Let $K$ be a finite extension of $F$. Let $[K : F]_i$ be the inseparable degree of $K/F$. Let $\bar{K}$ be an algebraic closure of $K$. Let $S$ be the set of $F$-embeddings ...
6
votes
3answers
156 views

Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
6
votes
2answers
206 views

$[L:K]=n!\ \Longrightarrow \ f$ is irreducible and $\text{Gal}(L/K)\cong S_n.$

How do I go about proving the following theorem ? Let $f\in K[T]$ have degree $n$ and splitting field $L/K$. Then we have $$ [L:K]=n!\ \Longrightarrow \ f\text{ is irreducible and Gal}(L/K)\cong ...
6
votes
1answer
138 views

Embedding of a field extension to another

Can $\mathbb{Q}(\sqrt {-2})$ be embedded into a cyclic extension of degree 4 over $\mathbb{Q}$?
6
votes
1answer
430 views

Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
6
votes
2answers
271 views

Field Isomorphisms

Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a ...
6
votes
1answer
777 views

Problem in Jacobson's Basic Algebra (Vol. I)

It is section $4.4$, exercise number $5$. It says the following: Let $F$ be a field of characteristic $p$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if ...
6
votes
2answers
375 views

Determining subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ without Galois theory?

Somehow I had convinced myself recently that one could determine the subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (here $\zeta_3 = \frac{-1+\sqrt{3}i}{2}$ is a primtive $3$rd root of $1$) without ...
6
votes
1answer
141 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 ...
6
votes
1answer
65 views

Is $\mathrm{GL}_n(K)$ divisible for an algebraically closed field $K?$

This is a follow-up question to this one. To reiterate the definition, a group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ ...
6
votes
2answers
373 views

A field that is an ordered field in two distinct ways

Question: Explain the construction below (taken directly from Counter Examples in Analysis): An ordered field is a field $F$ that contains a subset $P$ such that $P$ is closed with respect ...
6
votes
1answer
235 views

A characterization of finite purely inseparable extensions of fields

Let $K/k$ be a finite extension of fields. Let $A=K \otimes_k K$. An exercise: Show that $K/k$ is purely inseparable $\Leftrightarrow A/J(A) \cong k$, where $J(A)$ is the Jacobson radical of $A$. It ...
6
votes
3answers
188 views

Polynomials over finite fields

I’ve come across this problem in a coding theory course, and neither I nor several of my colleagues could solve it to our satisfaction. Let $F:=\mathrm{GF}\left(q\right)$ denote the field with $q$ ...
6
votes
3answers
77 views

Find minimal polynomial of this element?

Let $f(x)=x^3+x+1$. $\alpha_1, \alpha_2, \alpha_3 - $ roots of $f$. The task is to determine the minimal polynomial of $\frac{\alpha_1}{\alpha_2}$ over $\Bbb Q $ and $\Bbb Q(\alpha_1)$. My thoughts ...
6
votes
1answer
141 views

Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
6
votes
2answers
279 views

degree of a field extension

Let $\alpha$ be a root of $x^3+3x-1$ and $\beta$ be a root of $x^3-x+2$. What is the degree of $\mathbb{Q}(\alpha^2+\beta)$ over $\mathbb{Q}$? My guess is 9, because i found a monic polynomial of ...
6
votes
4answers
197 views

Number fields with all degrees equal to a power of three

Say that a number field $\mathbb K$ is $3$-powerful if the degree (over $\mathbb Q$) of every non-rational element of ${\mathbb K}$ is a power of $3$. By Zorn’s lemma, the field $\cal A$ of all ...
6
votes
1answer
133 views

Cyclic Extensions of $\mathbb{R}(t)$

Let $\mathbb{R}(t)$ be the field of rational functions over $\mathbb{R}$ (the fraction field of $\mathbb{R}[x]$). I am looking for elements in the Brauer group of the field, and the current idea I ...
6
votes
2answers
138 views

Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?

It is not hard to prove that every element of $\mathbb{F}_p$ has a square root in $\mathbb{F}_{p^2}$: take any $a \in \mathbb{F}_p$ and consider the polynomial $f = X^2 - a$. If $f$ has a root in ...
6
votes
1answer
189 views

Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?

Suppose that $L/K$ is a finite separable extension of fields and let $M$ denote the Galois closure of $L$. Let $\textrm{Hom}_K(L,M)$ denote the set of all $K$ - algebra homomorphisms from $L$ to $M$. ...
6
votes
1answer
646 views

Elementary field theory, field extensions of the rationals of degree 2

I've just started some reading and doing exercises on field theory with Galois theory in scope, and have had some trouble with this exercise. I think I have simply misunderstood some of the ...
6
votes
2answers
307 views

Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
6
votes
1answer
137 views

What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?

Just a small notation question from this Wikipedia page: The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$ ...
6
votes
1answer
374 views

Complete ordered field

I'm trying to prove that; If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F. Let A be a nonempty subset of F that is not a ...
6
votes
1answer
250 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
6
votes
2answers
275 views

Why is this extension of Galois?

Let $F$ be a subextension of $\mathbb{C}$ maximal with respect to not containing $\sqrt2$. Let $K/F$ be a finite extension with $K\subset\mathbb{C}$. Then $K/F$ is of Galois and $[K:F]$ is a power of ...
6
votes
3answers
158 views

almost n-th power in a field

The field of real numbers has a nice property - for a fixed $n\in \mathbb{N}$, every $r\in \mathbb{R}$ is either an n-th power (if n is odd or if r>0) or -r is an n-th power. I want to generalize this ...