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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255 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
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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, ...
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2answers
278 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
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1answer
740 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. ...
4
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1answer
944 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, $\Fract(F)= \{ {a \over b } \mid a \in F , b \in F \setminus \{ 0 \} \}$. ...
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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 ...
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3answers
318 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 ...
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3answers
343 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
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3answers
799 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 $F(a)=F(a^2)$?
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2answers
914 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 ...
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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 ...
5
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2answers
267 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
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1answer
243 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
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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
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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
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1answer
210 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 ...
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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 ...
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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 ...
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1answer
133 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 $f(x)$?
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1answer
198 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 ...
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1answer
899 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 ...
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3answers
523 views

On the meaning of being algebraically closed

The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$. By algebraic closure, every nonconstant ...
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1answer
946 views

Why is it called a 'ring', why is it called a 'field'?

The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers): addition is commutative $( 1 + 2 = 2 + 1 )$ addition and multiplication are associative $(2 +(2+2)) = ...
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3answers
2k views

Every algebraic extension of a perfect field is separable and perfect

I am trying to prove this statement in the characteristic $p>0$ case. Every algebraic extension of a perfect field is separable and perfect. This is stated as a corollary of Proposition ...
2
votes
2answers
169 views

When is this set a field?

For which $c$ will the set $K(c) = \{a + b\sqrt{c} : a, b \in \mathbb{Q}\}$ be a field? I know for example, that $K(\frac{2}{3})$ will be one, I am just wondering what the most general result of this ...
3
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1answer
202 views

What's a structure where $a-a = 1$ and $a\cdot a^{-1} = 0$?

I'm trying to specify a structure that has the basic features of a Boolean algebra but not necessarily restricted to binary sets. However, I observed that I almost have a field except that ...
2
votes
1answer
192 views

Does product of Galois groups equal to the Galois group of corresponding fields intersection?

Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=Gal(\bar{k}/k)$ be an Abelian group(may be infinite). If $K,L$ are intermediate fields, denote $G_K=Gal(\bar{k}/K), G_L=Gal(\bar{k}/L)$. ...
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3answers
282 views

Eigenvalues of a linear operator over a K-vector space

The following question may be somewhat ill-posed due to a lack of experience in dealing with fields. Let $T$ be a linear operator over a finite dimensional complex vector space. If the only (complex) ...
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1answer
349 views

Does every field have a non-trivial Galois extension?

Clearly a field $F$ with no Galois extensions must have only non-separable elements in any extension (otherwise, take the minimal polynomial of some separable element over $F$ - its splitting field ...
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3answers
292 views

What properties of numbers allow us to remove parentheses from expressions?

I've seen it asserted in several places (e.g., Spivak's Calculus, p.3) that the fact that "parentheses can be freely rearranged" in expressions involving only addition ($+$) is based solely on (P1) ...
2
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1answer
75 views

If $L$ contains a $n$-th root of $a\in K$, why does $K$ already contain a $n$-th root?

I'm trying to solve the following problem from Algebra by Siegfried Bosch (english version below): Es seien $m,n$ teilerfremde positive ganze Zahlen. Ist dann $L/K$ eine Körpererweiterung vom Grad ...
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2answers
333 views

Field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$

Am I right to say that the field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$ is an algebraic extension of $\mathbb{Q}$? Because $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt{3})( ...
3
votes
1answer
278 views

Finite separable extension over fixed field $K=L^\sigma$ is normal?

I'm trying to prove the following: Let $L$ be an algebraically closed field and let $\sigma \in \mathrm{Aut}(L)$. I want to show that for any separable $\alpha \in L$ over $K = L^\sigma$ ($K$ is ...
3
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2answers
801 views

Galois group of $x^6+3$ over $\mathbb Q$

I'm having some difficulties finding the Galois group of the polynomial $g(x)=x^6+3$ over $\mathbb Q$. Here's what I did : I observed that the roots of the given polynomial are $\sqrt[6]3 ...
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1answer
178 views

A question about splitting field [duplicate]

Possible Duplicate: Splitting field of $x^{n}-1$ over $\mathbb{Q}$ Let $A$ denote the splitting field of $x^n-1$ over $\mathbb{Q}$, so what is the dimension of $A$ as a linear space over ...
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6answers
667 views

Is it possible to have a field without an additive identity?

If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization? By definition it is not possible, but are there ...
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2answers
2k views

Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field

I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I ...
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5answers
3k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
4
votes
1answer
2k views

the degree of a splitting field of a polynomial

Let $f(x)\in F[x]$ be a polynomial of degree $n$. Let $K$ be a splitting field of $f(x)$ over $F$. Then [K:F] must divides $n!$. I only know that $[K:F] \le n!$, but how can I show that $[K:F]$ ...
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0answers
73 views

Partial differential fields and logic

Is the theory of partial differential fields a first-order or higher-order logic theory? Is the theory of differential fields already equipped with partial derivatives? I was told that the General ...
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1answer
131 views

What's the index of a subfield?

When you talk about groups $[G:H]$ is the number of H-cosets in G. My book has recently started using this notation with fields, and I'm not sure what it means. My first thought was that you could ...
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1answer
118 views

Vector Spaces v.s. Fields

Vector spaces seem to have a very similar definition to fields. Are vector spaces fields? Thanks.
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1answer
254 views

Principle of substitution — fields

Could someone possibly explain this definition (applied to fields) to me? The principle of substitution: In a field F, we can, in any formula involving an element $\alpha\in F$, replace $\alpha$ ...
5
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1answer
899 views

Least upper bound property iff convergence of Cauchy sequences

From http://en.wikipedia.org/wiki/Non-Archimedean_ordered_field: The field of rational functions over $\mathbb{R}$ can be used to construct an ordered field which is complete (in the sense of ...
6
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1answer
304 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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1answer
117 views

Classification of field extensions

Is it true that all extension of field $k$ are subfields of $\bar k$ (algebraic closure of $k$)? The same question for all algebraic extension. Thanks.
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2answers
167 views

(The number of) embeddings of an algebraic extension of $\mathbb{Q}$ into $\mathbb{C}$

While reading a book (Neukirch) I came upon the claim that for an algebraic extension $K/\mathbb{Q}$ of degree $n$ there are $n$ embeddings of $K$ into $\mathbb{C}$. Why is it so, and in general how ...
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1answer
306 views

What happens when you mod out by a non-primitive irreducible polynomial over $F_q$?

What is the difference between modding out by a primitive polynomial and modding out by a non-primitive irreducible polynomial in a finite field $F_q$? From what I understand either one should ...
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4answers
883 views

Is $\mathbb{Q}[2^{1/3}]$ a field?

Is $\mathbb{Q}[2^{1/3}]=\{a+b2^{1/3}+c2^{2/3};a,b,c \in \mathbb{Q}\}$ a field? I have checked that $b2^{1/3}$ and $c2^{2/3}$ both have inverses, $\frac{2^{2/3}}{2b}$ and $\frac{2^{1/3}}{2c}$, ...