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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98 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
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1answer
33 views

How to show the cyclotomic polynomial is irreducible over $\mathbb{R}(T,\sqrt[n]{T})$

I'm trying to solve the following problem. Let $T$ be a transcendental over $\mathbb{R}$. Put $K=\mathbb{R}(T), n\geq3$. Let $L$ be the least splitting field of $X^n-T$. Then, calculate $[L:K]$. I ...
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2answers
42 views

prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.

let $R$ be a unique factorization domain and let $F$ be its field of fractions. Prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.
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1answer
42 views

How to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$

I'm trying to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$. But this is harder than I expected. Is there any easy way to show this?
2
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1answer
90 views

Ordered field with nested intervals axiom without Archimedean axiom

Does an ordered field $F$ satisfying the Cauchy-Cantor axiom but not the Archimedean axiom exist? Cauchy-Cantor axiom: Every system of nested intervals $I_1 \supset I_2 \supset \cdots \supset ...
5
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2answers
181 views

A field having an automorphism of order 2

The following fact is used in the Unitary space. If $F$ is a field having an automorphism $\alpha$ of order 2. Let $F_0=\{a\in F: \alpha(a)=a\}$. Then $|F:F_0|=2$. Is there any easy proof (or ...
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3answers
136 views

Show that these matrices are congruent.

Let $K$ be a field of characteristic$\ne 2$ and $u$ be an invertible element of $K$. Show that $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\begin{pmatrix}u&0\\0&-u\end{pmatrix}$ are ...
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3answers
99 views

Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
3
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1answer
72 views

Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
2
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3answers
239 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
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1answer
69 views

Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...
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2answers
365 views

Are all existence proofs by contradiction?

Do all existence proofs proceed indirectly (a.k.a "by contradiction," "reductio ad absurdum," etc.)? For example, can we prove directly that in a field the additive identity element and the ...
1
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1answer
38 views

Degree of a single extension of a field. [closed]

Let $K$ be a field. Let $L:=K(b)$ where $b^m\in K$ but $b^i\not\in K$ for all $0<i<m$. Then is $[L:K]=m$?
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1answer
64 views

Characterize elements in $\mathbb Q(\sqrt2,\sqrt[3]5)$

I'm studying algebra by Herstein's Topics in Algebra, 2ed, and stuck at Ex.5.1.6(b): In $\mathbb Q(\sqrt2,\sqrt[3]5)$ characterize all the elements $w$ such that $\mathbb Q(w)\ne \mathbb ...
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2answers
41 views

Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective?

Why is $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism given $\phi : F[x] \rightarrow F(a)$ satisfy $\ker \phi = \{0\}$ ? I've been trying to figure out why $\bar \phi$ is an isomorphism, and ...
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1answer
53 views

How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
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2answers
49 views

Mathematical field theory Application in real world and other branches of Mathematics

I need to write work about applications of Mathematical field theory in real world and in other branches of Mathematics. Can someone guide me to an appropriate book, resource? Thanks, Denis.
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1answer
92 views

Radical Solvable quintic polynomial.

I am trying to solve the following exercise: Let $k$ be a field of characteristic 0, and let $f(x)\in k[x]$ be a polynomial of degree 5 with splitting field $E/k$. Prove that $f(x)$ is ...
4
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1answer
80 views

What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
2
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1answer
45 views

$(u+1)(u+2)\cdots(u+p)=u^p-u$

Let $E$ be a field with characteristic $p>0$. How should I prove that $$(u+1)(u+2)\cdots(u+p)=u^p-u$$ I can verify this equality for small $p=2,3,5$. Is there a way to prove this result in general? ...
3
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3answers
225 views

What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...
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1answer
250 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
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1answer
37 views

A question about separable extension

We say $K$ is separably generated over $k$ if there exists a transcendence basis $\{x_i,i\in I\}$ of $K/k$ such that $K/k(x_i,i\in I)$ is a separable algebraic extension. We say $K$ is separable over ...
5
votes
1answer
54 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
3
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0answers
38 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
3
votes
2answers
54 views

Is there a universal property in this proposition (which regards field extensions)?

Since learning a bit of category theory, I am trying, as an exercise, to state results that I come across in categorical language. I am trying to do this with the following: Let ...
3
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1answer
46 views

what is this property called?(Field theory)

In what references the following property $P$ of a field $F$ is investigated? The property $P$: For all $n\in \mathbb{N}$ if $\sum_{i=1}^{n} f_{i}^{2}=0,\;\;f_{i}\in F$ then $f_{i}=0, \forall i\in ...
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2answers
90 views

Euclid and finite fields

In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set. Much ...
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1answer
48 views

Possible Fields?

Is there an algebraically closed field which is a 1-dimensional vector space (as opposed to complex numbers which are 2-D)? Also is there a complete $\aleph_0$ field?
2
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1answer
29 views

If $L_1/K$ and $L_2/K$ are not Galois (solvable), then $L_1L_2/K$ is not Galois (solvable)

This is part of an exam preparation: Prove/contradict: If $L_1/K$ and $L_2/K$ are not Galois, then $L_1L_2/K$ is not Galois. If $L_1/K$ and $L_2/K$ are not solvable Galois extensions, ...
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2answers
32 views

Is $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$

Is $$\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$$ I know that $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}(i)$ but I can't say that $\mathbb{Q}[X]/(X^2-1) \cong \mathbb{Q}(1) = \mathbb{Q}$ ...
2
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0answers
35 views

Why is the subfield (of a field) generated by an algebraic element equal to the subring generated by the same element?

I am trying to prove that, for a field extension $\mathbf{K}/k$ and $a$ an algebraic element over $k$, $$k(a)=k[a],$$ where $k(a)$ is the subfield of $\mathbf{K}$ generated by $a$ and $k[a]$ is the ...
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1answer
40 views

A field extension of degree 8

I would really appreciate it if you give me a hint on the following question: If $K \subset F$ is a field extension of degree 8, then we must have $F=K(a,b,c)$ for some a, b and c in F.
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2answers
87 views

Dimension of $\Bbb Q(e)$ over $\Bbb Q$?

The dimension of $\Bbb Q(\sqrt{2})$ over $\Bbb Q$ is finite since $\sqrt2$ is algebraic over $\Bbb Q$. But what about any transcendental number (say $e$)? Which is the smallest field containing $\Bbb ...
2
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1answer
24 views

Field Isomorphisms between a field and something that contains it

Are there any k-isomorphism of fields between M and L such that K $\subseteq$ M $\subset$ L? Examples would be appreciated. Thanks
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6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
2
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2answers
37 views

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
5
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2answers
392 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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1answer
38 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
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1answer
26 views

Normalizer of a subgroup of a Galois group

I wanted to check whether my solution for this problem was correct. Let $k \subseteq L \subseteq K$ be a finite extension of fields, with $K/k$ Galois $H$ the normalizer of $Aut(K/L)$ in $Aut(K/k)$. ...
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1answer
36 views

A request for a particular example in field theory.

I'm looking for an example of the following kind: Let $a,b\notin \Bbb{Q}$, where $a$ and $b$ satisfy the irreducible polynomials $p(x)$ and $q(x)\in\Bbb{Q}[x]$ respectively. The irreducible ...
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votes
2answers
319 views

Is there a 'conjugation' on every algebraically closed field?

Let $K$ be an algebraically closed field. Then the polynomial $x^2+1\in K[x]$ has two distinct roots (when $K$ doesn't have characteristic 2). Let's suggestively call them $i$ and $-i$. Does there ...
0
votes
0answers
23 views

Hensels Lemma in many variables

Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$. Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable ...
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2answers
44 views

Subgroup of roots of unity of a field. [closed]

Let $F$ be a field. Show that the set of all $n$th roots of $1$ is a subgroup of $F^\times$.
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1answer
92 views

What is the left adjoint of the forgetful functor from fields to integral domains?

I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain: "There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category ...
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1answer
48 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
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1answer
59 views

If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
0
votes
1answer
57 views

Prove $Z_{p}$ are prime fields,where $p$ is prime numbers

show that $Z_{p}$ are prime fields,where $p$ is prime numbers. maybe this problem is old,But I look for some book,and can't find it,someone know which book have this problem proof? because I know ...
1
vote
1answer
59 views

Proof that $a\equiv b \pmod n \iff a \pmod n = b\pmod n$

Proof that for every $a,b \in \mathbb Z,\ n \in \mathbb N$, that $$a\equiv b \pmod n \iff a \pmod n = b \pmod n.$$ My approach is: $n\mid a$ and $n\mid b$ $a\equiv b \pmod n \iff \exists x,y: ...
3
votes
2answers
33 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...