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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Compositum of fields with trivial intersection

Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
7
votes
1answer
117 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
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1answer
150 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 ...
7
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594 views

How to find irreducible polynomials over $\mathbb{Q}(i)$ with prescribed Galois group?

Here is my recent homework question: For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example ...
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1answer
180 views

Calculating Separable Closures

In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion ...
7
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1answer
88 views

Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
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2answers
247 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
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1answer
108 views

Multivariate coprime polynomials in field extensions

Suppose $f$ and $g$ are polynomials in $n$ variables, $n\ge 2$, over a field $E$. Suppose further that $f$ and $g$ are relatively prime over $E$. If $F$ is a field extension of $E$, are $f$ and $g$ ...
7
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1answer
197 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
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1answer
222 views

Two questions on Nagata's counterexample to the Hilbert's fourteenth problem.

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
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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 $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?
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1answer
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Extension of Homomorphisms (Lang, Atiyah and McDonald)

Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the ...
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1answer
258 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 ...
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1answer
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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 $K$. ...
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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 ...
7
votes
0answers
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Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
6
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4answers
593 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.
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3answers
458 views

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

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

Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

$F$ is a field of order $32$. $F$ and {$0,1$} are trivial subfields of $F$. But how can we show that these are the only subfields of $F$? Can someone give me a direction to this question?
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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 $n$ ...
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3answers
226 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?
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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 $$ ...
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5answers
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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?
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701 views

Why are fields with characteristic 2 so pathological?

For example, over fields with characteristic 2, there exist nonzero symmetric nilpotent matrices, and nonzero matrices could be simultaneously symmetric and anti-symmetric. I wonder why characteristic ...
6
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2answers
376 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
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1answer
336 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' ...
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3answers
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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 ...
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5answers
145 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 $(a,b)(c,d)=(ac-db,ad+bc)$ ...
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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 ...
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171 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?
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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
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4answers
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If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
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5answers
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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 ...
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2answers
1k 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.
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3answers
689 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 ...
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175 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
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2answers
211 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
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2answers
295 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 ...
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1answer
144 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
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1answer
363 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
6
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1answer
537 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 ...
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2answers
407 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 ...
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1answer
143 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 ...
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votes
3answers
226 views

Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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3answers
891 views

Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$

Determine all the subfields of the splitting fields of this polynomial. I chose this problem because I think to complete it in great detail will be a great study tool for all of the last chapter, as ...
6
votes
1answer
66 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 $g=h^k.$ ...
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2answers
406 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 ...
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1answer
242 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 ...
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1answer
291 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
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2answers
994 views

Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8 Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...