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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13
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3answers
464 views

Why is the difference of distinct roots of irreducible $f(x)\in\mathbb{Q}[x]$ never rational?

The way I understand it, is that if $f(x)$ is an irreducible polynomial in $\mathbb{Q}[x]$ of degree at least 2, then a difference of distinct roots $a_i-a_j$ is never rational for any of the ...
15
votes
2answers
1k views

Basis of primitive nth Roots in a Cyclotomic Extension?

While reading one of Keith Conrad's great blurbs, Linear Independence of Characters, there is a footnote at the bottom of page 2 saying In general, the primitive $n$th roots of unity in the $n$th ...
3
votes
3answers
397 views

Infinite Field of Characteristic 5

I recently took an exam in which the professor asked to give an example of an infinite field of characteristic 5. I had studied this problem, and found examples such as this. My answer that I ...
2
votes
1answer
495 views

Finding some missing subfields of a splitting field of $x^4-7$

I was looking at the Galois group of the splitting field of $x^4-7$ over $\mathbb{Q}$. I found it to be $\mathbb{Q}(\sqrt[4]{7},i)$, and the Galois group to be the dihedral group of order $8$. Now ...
10
votes
2answers
868 views

What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?

Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$. So the ...
9
votes
2answers
2k views

Why does an irreducible polynomial split into irreducible factors of equal degree over a Galois extension?

I've been struggling to prove this fact over the past day or so. Suppose $f(x)\in F[X]$ is irreducible over a field $F$ with $\deg(f)=n$, and let $L$ be the splitting field of $f(x)$ over $F$ with ...
3
votes
1answer
859 views

Characteristic of a field is $0$ or prime

I'm trying to prove that the characteristic of any field $F$ is either $0$ or a prime number, but I have no idea what to do. Help?
2
votes
3answers
473 views

A ring with a subring that is a field

I'm looking for an example of a ring $R$ such that $R$ has no multiplicative identity, but R has a subring $A$ which is a field
1
vote
2answers
438 views

Field of Quotients of an integral domain may also be a field of quotients?

I'm trying to show by an example that a field $F$ of quotients of a proper subdomain $A$ of an integral domain $D$ may also be a field of quotients of $D$. I have no idea where to begin. Help?
3
votes
1answer
906 views

Question about field norm from Dummit and Foote 14.2.17(d)

I'm trying to solve the last part of an exercise in Dummit and Foote. Let $K/F$ be any finite separable extension, and let $\alpha\in K$. Let $L$ be a Galois extension of $F$ containing $K$ and ...
1
vote
2answers
343 views

Normal extension of any degree

Could any one tell me how to show that for any positive integer n, there exists a normal extension of rational number field of degree n?
0
votes
1answer
87 views

Definition of polynomial solutions over polynomial field $F[x]$

I'm studying abstract algebra and ran into the problem of solving equations where solutions are polynomials. The problem is as follows: Given B a member of a polynomial field $F[x]$, having ...
29
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
12
votes
2answers
224 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
2
votes
2answers
156 views

Chain of fields that are Galois over all subfields

Is there an example of fields $F_1$, $F_2$, and $F_3$ such that $\mathbb{Q}\subset F_1\subset F_2\subset F_3$ such that $[F_3:\mathbb{Q}]=8$ and each field is Galois over all its subfields but $F_2$ ...
3
votes
1answer
893 views

Fastest way to compute subfields of $\mathbb{Q}(\sqrt[8]{2},i)$ which are Galois over $\mathbb{Q}$?

I have the lattice of subfields of the splitting field $\mathbb{Q}(\sqrt[8]{2},i)$ over $x^8-2$, and the corresponding lattice of subgroups of the Galois group $G$ of the splitting field. I'm now ...
4
votes
3answers
630 views

GCD in polynomial rings with coefficients in a field extension

Let $E|F $ be a field extension and $f,g$ $\in$ $F[x]$ (the polynomial ring with coefficients in $F$ ). Let's denote with $(f,g)_F$ the greatest common divisor of $f$ and $g$ in $F[x]$. Is it true ...
3
votes
1answer
126 views

On Intermediate Fields of $\mathbb{C}(x_1,\dots,x_n)$

I am recently reading some Galois Theory, and a question occurred to me: What are the intermediate fields of $K$ of $\mathbb C(x_1,\dots,x_n)$, where $n$ is an arbitrary integer? I am aware of a ...
5
votes
2answers
215 views

If $X^n+Y^n+1$ is reducible, is the degree divisible by the characteristic?

I was playing around with the Frobenius map, and made a small observation. Suppose $F$ is a field, and $F[X,Y]$ is the corresponding polynomial ring in two indeterminates. If $\text{char}(F)=p$ ...
8
votes
2answers
177 views

Intermediate field between $F$ and $F(x)$

Suppose that $F$ is a field and that $u \in F(x):= \{PQ^{-1}:P,Q \in F[x], Q\neq 0 \}$, so that $F \subseteq F(u) \subseteq F(x)$. Is there a general method for determining $[F(x):F(u)]$? For my ...
2
votes
1answer
870 views

Order of Galois group divides the degree of the extension

I keep seeing this theorem used in many textbooks but none of them provide proof (or there is no text layer so I can't find it!). Here is the statement in Algebra (Artin, pg. 540): (1.6) Theorem. ...
1
vote
1answer
153 views

Polynomials in several variables over a field

I am very new to field theory and I am trying to prove that if $F$ is a field and $R \in F(x_1,x_2,\ldots,x_n):=\{ PQ^{-1} : P,Q \in F[x_1,x_2,\ldots,x_n] \}$ is nonconstant, then $R$ is ...
2
votes
1answer
227 views

Normalised absolute values on $p$-adic extensions

I have the following problem: show that if $L/K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$ with normalised absolute values $|\cdot|_K$ and $|\cdot|_L$, with $n=[L:K]$, then ...
1
vote
1answer
91 views

When does the fraction field of a ring have a non-trivial Galois extension

I have read this previous question on existence of a non-trivial Galois extension. I was wondering about the following situation. Suppose, $R$ is a domain that is not a field. When does the fraction ...
23
votes
5answers
4k views

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...
6
votes
2answers
2k views

How do I find a splitting field $x^8-3$ over $\mathbb{Q}$?

Here's the situation. I am in this algebra class, and so far we have defined splitting fields and proved their existence and uniqueness. We have not yet decided on any rigorous definition of complex ...
3
votes
2answers
191 views

Splitting of primes in Galois extensions

I have the following problem: suppose $F/K$ is an abelian (Galois) extension of number fields, Galois group G, and $\mathfrak{p}$ is a prime of K, $\mathfrak{P}$ a prime of F dividing $\mathfrak{p}$; ...
8
votes
2answers
521 views

Is $\sqrt{2}\in\mathbb{Q}(\sqrt[8]{3})$ or not?

My hunch is that $\sqrt{2}\not\in\mathbb{Q}(\sqrt[8]{3})$. For practice, I want to compute the splitting field and its degree of $x^8-3$ over $\mathbb{Q}$. I know the roots are ...
1
vote
2answers
334 views

Is there an explicit description of the fields laws on this field?

I'm working on a fairly simple problem about a field, but I want to know if the operations can be explicitly described. Suppose $c$ is not a quadratic residue modulo $p$, and consider the quotient ...
2
votes
0answers
108 views

Quick question: finite extensions and norms

[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."] I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
8
votes
2answers
289 views

Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
4
votes
1answer
819 views

finding the multiplicative inverse in a field

Let $L/K$ be a field extension. Let $a\in L$ and $K[a]=\{p(a)\;|\; p\in K[x]\}$; then $K[a]$ is clearly an integral domain. I want to show that when $a$ is algebraic over $K$, then $K[a]$ is a field. ...
5
votes
1answer
1k views

The field of fractions of a field $F$ is isomorphic to $F$

Let $F$ be a field and let $\newcommand{\Fract}{\operatorname{Fract}}$ $\Fract(F)$ be the field of fractions of $F$; that is, $\Fract(F)= \{ {a \over b } \mid a \in F , b \in F \setminus \{ 0 \} \}$. ...
0
votes
1answer
97 views

Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$?

Let $M$ be a finite extension of $E$ and let $E$ be a finite extension of $F$. Then $M$ is a finite extension of $F$ and $[M:F] = [M:E][E:F]$. Is there an easy explanation and/or proof for this ...
10
votes
3answers
353 views

What kinds of non-zero characteristic fields exist?

There are these finite fields of characteristic $p$ , namely $\mathbb{F}_{p^n}$ for any $n>1$ and there is the algebraic closure $\bar{\mathbb{F}_p}$. The only other fields of non-zero ...
5
votes
3answers
345 views

Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$?

Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$ when $\mathbb{Q}(\alpha)$ is a $p$th degree Galois extension of $\mathbb{Q}$? ($p$ is prime) I got stuck with this ...
2
votes
3answers
892 views

Equal simple field extensions?

I have a question about simple field extensions. For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$?
1
vote
2answers
1k views

characteristic of a finite field

knowing that the characteristic of an integral domain is $0$ or a prime number, i want to prove that the characteristic of a finite field $F_n$ is a prime number, that is $\operatorname{char}(F_n)\not ...
9
votes
2answers
2k views

reducible polynomial modulo every prime

how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$. For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
6
votes
2answers
279 views

Is $\mathbb R$ terminal among Archimedean fields?

I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is: Because every Archimedean ordered ...
2
votes
1answer
295 views

Interpretation of a question: “group of all p-power roots of unity”

I have a homework problem I'm trying to do, but I'm not sure what it's asking. The problem is as follows: Recall that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the group of all roots of unity in ...
6
votes
1answer
129 views

For which $k$ do the $k$th powers of the roots of a polynomial give a basis for a number field?

Let $f \in \mathbb{Q}[x]$ of degreee $d$ be irreducible, with roots $\alpha_1,\ldots, \alpha_d$. One particular basis for the field extension of $\mathbb{Q}$ obtained by adjoining the roots of $f$ is ...
2
votes
3answers
110 views

Show $xy\neq0$ is the same as $x\neq0 \wedge y \neq0$

I have to show: $$xy\neq0 \Leftrightarrow x\neq0 \wedge y \neq0 $$ I think I can "simplify" it to this: $$xy=0 \Leftrightarrow x=0 \vee y=0 $$ Since $a\cdot0=0$ is an proven theorem, I can show: ...
2
votes
1answer
217 views

Finding Polynomials of Intermediate Galois extensions

Let $G$ be the Galois group of an irreducible polynomials $f(x)$ in $\mathbb{Q}[x]$. Let $K$ be the splitting field of $f(x)$. From the fundamental theorem of Galois theory we have that the ...
2
votes
2answers
1k views

What is Galois Field

When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in ...
1
vote
3answers
82 views

The intersection of $O_K$ with $K^\ast$

Let $K/\mathbf{Q}$ be a number field with ring of integers $O_K$. Is $O_K\cap K^\ast = O_K^\ast$? I can't show that the inverse of an element in $O_K\cap K^\ast$ lies in $O_K^\ast$...
2
votes
1answer
100 views

Is it always true that |1+1|>1 in an Archimedean valuated field?

The following is a sentence from the proof of the theorem 1.2 (P.14-15) in Topological Vector Spaces (Third Printing Corrected 1971) by H.H.Schaefer Finally, if $K$ is an Archimedean valuated ...
0
votes
1answer
134 views

Minimal Polynomial over different fields

Let $K \subset L \subset M$. Let $f(x)$ be the minimal polynomial for $\alpha$ over $M$. Moreover, suppose $f(x)\in L[x]$, then is the minimal polynomial for $\alpha$ over $L$ also $f(x)$?
4
votes
1answer
225 views

Dimension of compositum of field extensions

Suppose we have fields $L$, $M$ and $N$ all infinite algebraic Galois extensions of a field $k$ such that $L \cap M$, $L \cap N$ and $N \cap M$ are finite dimensional extensions of $k$. Then is $L ...
6
votes
1answer
997 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...