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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1answer
36 views

How do we know that $\mathbb{Q}$ is the initial field of characteristic $0$?

For any field $F$, I can see why any two morphisms $\mathbb{Q} \rightarrow F$ must be equal. If $F$ has characteristic $0$, how do we furthermore know that there exists a homomorphism $\mathbb{Q} ...
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3answers
43 views

For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?

sorry if this question seems obvious, For a field F and $a,b\notin F$, does $[F(a,b):F(a)]=[F(b):F]$? If so, how do you prove it or is there a counter example?
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2answers
85 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
18 views

Is a finitely generated extension of a real closed field also real closed?

Let $\widetilde{\mathbb{Q}}$ be the field of real algebraic numbers, and consider $\widetilde{\mathbb{Q}}(\pi)$. My question: is $\widetilde{\mathbb{Q}}(\pi)$ a real closed field? Bonus karma points ...
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0answers
38 views

Proof that all ordered fields are in the Surreals

It says on Wikipedia that any ordered field can be embedded in the Surreal number system. Is this true? How is it done, or if it is unknown (or unknowable) what is the proof that an embedding exists ...
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1answer
32 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
88 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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0answers
25 views

Schanuel's conjecture and field extensions.

Doing a little bit of reading over the summer break before going into my masters year of my maths degree and i have been looking at Schanuel's conjecture which states that; Given any $n$ complex ...
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2answers
40 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
40 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?
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1answer
80 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 ...
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votes
2answers
177 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
92 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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2answers
1k 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 ...
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3answers
131 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 ...
3
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1answer
68 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 ...
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2answers
906 views

Is vector space a field? Or more than that?

As you all know, vector space is closed under scalar multiplication, scalar product, vector product and addition. If I take scalar product, vector space is a field, but if i take vector product, ...
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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
59 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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3answers
223 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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2answers
363 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 ...
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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
40 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
49 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 ...
6
votes
1answer
250 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
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2answers
42 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.
3
votes
3answers
222 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 ...
2
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1answer
86 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 ...
3
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1answer
70 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 ...
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2answers
222 views

Transcendental extension that is not simple

Let $K$ be a field and $x, y$ be independent variables. How can I show that $K(x, y)/K$ is not a simple extension?
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1answer
44 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? ...
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0answers
43 views

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
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1answer
544 views

Field Automorphisms

Quite a general question: For a subfield $K$ of $\mathbb{C}$, how can we prove that every automorphism of $K$ fixes every rational number, and secondly, that the set of such automorphisms forms a ...
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1answer
34 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 ...
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1answer
235 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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2answers
46 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 ...
5
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1answer
52 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 ...
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0answers
35 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 $K$ ...
2
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1answer
27 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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6answers
1k 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 ...
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1answer
45 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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1answer
47 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?
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2answers
280 views

Specific way of showing $\Bbb Z[\sqrt{-d}]$ is not a Euclidean Domain when $d>2$

Is it true that if a ring is not a UFD then it's not a Euclidean Domain? I have a ring $R=\mathbb{Z}[\sqrt{-d}]=\{ a+b\sqrt{-d} \mid a,b \in \mathbb{Z} \}$ where $d$ is a square free integer. I want ...
3
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2answers
114 views

Galois Group for $x^5-1$

This question is an extension to the question in math.stackexchange.com/questions/759230/subfield-of-the-galois-group-of-x5-1 It seems the discussion in that topic is dead and I still have a major ...
3
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1answer
48 views

Is $\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$ a simple extension?

Is the extension $$\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$$ simple? If so find the minimal polynomial and the basis for the extension.
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2answers
31 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 $\mathbb{Q}[X]/(X^2-1) \cong \mathbb{Q}(1) = \mathbb{Q}$ ...
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1answer
103 views

Algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
3
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0answers
26 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 ...
3
votes
2answers
32 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 ...
1
vote
1answer
39 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.