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Questions tagged [field-theory]

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 topic.

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1answer
22 views

Complex conjugate and Field Extension

Let $E$ be a subfield of $\mathbb{C}$ and Let $\overline{E}=\{\overline{z} \, |\, z \in E \}$ with $\overline{z}$ being the complex conjugate of $z$. Let $K$ be a subfield of $\mathbb{C}$ with $\...
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1answer
9 views

How to show that this simple extension avoids adding more elements in $L$?

$F\subset L \subset K$ is a field extension, $α \in K$ is algebraic element over $F$ , whose minimal polynomial $p(x)$ over $F$ is irreducible over $L$, show that $F(α)\cap L = F$ I don't know how to ...
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0answers
40 views

Does there exist an automorphism of $\mathbb{C}$ that does not keep $\mathbb{R}$ fixed? [duplicate]

I'm going to take Galois theory in the upcoming semester and I'm working on some basic problems right now. I was trying to figure out what $\mathrm{Aut}(\mathbb{C})$ should be. If $f: \mathbb{C} \to \...
2
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3answers
58 views

Proof verification: Proving or disproving that $(\Bbb R^+, \oplus, \otimes)$ as defined is a field

The problem I have been working on, in summary: Let $R$ denote the set of positive real numbers. Define addition, denoted $\oplus$, and multiplication, dentoed $\otimes$, respectively by $a \oplus ...
2
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1answer
44 views

Is this a counterexample?

Suppose $K $ is a field and $\overline K $ an algebraic closure. Let $f $ be a $K $-automorphism of $\overline K$, let $L$ be the subfield of $\overline K $ fixed by $f $. In this post : (link), they ...
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0answers
19 views

Algebraic Field Extensions and Irreducible Polynomials

Let $E/K$ be a field extension, $a,b\in E$ algebraic over $K$. Show: $\text{Min}(a,K,X)$ irreducible over $K(b)$ if and only if $\text{Min}(b,K,X)$ irreducible over $K(a)$ My attemp: (1) Let deg $\...
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1answer
47 views

find all algebraic elements of $\mathbb{Q}(\pi)$ over $\mathbb{Q}(\pi^2 -2\pi +5)$

Well, it is pretty clear that $\mathbb{Q}(\pi^2-2\pi+5)$ are all algebraic. I don't have any idea how to justify it, but I think those are the only algebraic elements of $\mathbb{Q}(\pi)$ over $\...
2
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1answer
26 views

Field extensions and Intermediate Rings

Let $E/K$ be a field extension. Show the following equivalence: (i) E/K is algebraic (ii) Every Intermediate Ring of $E/K$ is a Field. Here is what I tried: (i)$\Rightarrow$(ii) Let $E/K$ be ...
4
votes
4answers
58 views

Showing existence of irreducible polynomial of degree 3 in $\mathbb{F}_p$

I'am trying to show that for every p$ \in \mathbb{N}$ where p is prime, there is an irreducible polynomial of degree 3 in $\mathbb{F}_p$. I've found too general answers for that question, but I want ...
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0answers
27 views

Algebraic Closure - equivalent definitions [on hold]

I want to show that the following characterizations of the algebraic closure are equivalent. Let K be a field (i) $E=K$ for every algebraic extension $E/K$ (ii) Every non-constant Polynomial in $K[...
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2answers
68 views

What is the minimal polynomial of $\sqrt{3+4i}+\sqrt{3-4i}$ over $\mathbb{Q}$?

My attempt: I know that $P=\mathbb{Q(\sqrt{3+4i}+\sqrt{3-4i})} \subseteq \mathbb{Q(\sqrt{3+4i},\sqrt{3-4i})}=K$. But I don't know whether $K \subseteq P$. Assuming $K=P$. I can see that K is Galois ...
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0answers
49 views

How to find a basis of $\Bbb R(t)$ over $\Bbb R$?

I am reading field theory. I dont understand the following fact: What will be a basis of $\Bbb R(t)=\{\frac{f(t)}{g(t)}:f(t),g(t)\in \Bbb R[t]\}$. I know that $\Bbb R(t)$ is a field and hence a ...
2
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1answer
21 views

Example of an infinite simple extension.

I'm new to Field Theory and I'm looking for an example of an infinite simple extension. Theorem: The element $\alpha$ is algebraic over $F$ if and only if the simple extension $F(\alpha)/F$ is ...
2
votes
2answers
72 views

Why must an automorphism of an extension of $\mathbb{Q}$ send 1 to a rational number?

This is from a video I was watching that claimed this: If $\phi$ is an automorphism of an extension field $F$ of $\mathbb{Q}$, then $\phi(q)=q$ for all $q\in\mathbb{Q}$. The proof started by ...
5
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2answers
95 views

Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$

I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
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0answers
25 views

Compute the Galois group of L over K [duplicate]

$\textbf{Problem}$ $K=\mathbb{Z}/p\mathbb{Z}$; L=splitting field of $\prod_{i=1}^{p-1} (t^2-i)$, where $p$ is an odd prime $\textbf{Attempt}$ Since $(-k)^2=(p-k)^2=k^2$, we know that there are $\...
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1answer
19 views

Degree of splitting fields

I'm learning about splitting fields but I'm not sure if I am right. Hopefully I can get some insights on whether I have been learning correctly. The question asks to find the degree of the splitting ...
1
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2answers
48 views

Describe $\Bbb{R}[x]/(x^2 + 1)$

First by previous knowledge, I do know that $\Bbb{R}[x]/(x^2 + 1) \cong \Bbb{C}$, so this might seem trivial. But I am not here to ask about that and I don't want to use this fact I am reading Dummit ...
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0answers
3 views

Question regarding separable extension

If $K$ is a field extension of $F$ and if $\alpha$ $\in$ K is not separable over F, show that $\alpha^{p^{m}}$ is separable over F for some $m\geq 0$. Where $p=char(F)$. I have been trying it for ...
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0answers
34 views

Number Fields with Cyclic Galois Groups

Let $\xi\in\overline{\mathbb{Q}}\backslash\mathbb{Q}$ be an algebraic number of degree $d\geq2$ and let $\mathbb{K}=\mathbb{Q}\left(\xi\right)$. What are some conditions on $\xi$ and $d$ ...
4
votes
2answers
104 views

Turning $\mathbb R^n$ into field

I am reading Apostol's fascinating text Mathematical Analysis. In a footnote on P117, he writes: If it were possible to define multiplication in $\mathbb R^3$ so as to make $\mathbb R^3$ a field ...
3
votes
1answer
31 views

Let K be the splitting field of the polynomial $x^4-2x^2-2$. Find an automorphism $\sigma \in Gal(K/\mathbb{Q})$ of order 4.

My attempt: The roots of the polynomials are $x=\sqrt{1+\sqrt{3}}, -\sqrt{1+\sqrt{3}}, -\sqrt{1-\sqrt{3}}, -\sqrt{1-\sqrt{3}}$. and the automorphisms $\sigma \in Gal(K/\mathbb{Q})$ permutes the root ...
1
vote
1answer
58 views

Major misunderstanding about field extensions and transcendence degree

So presumably this question is very basic, but I'm having some trouble with apparent contradictions in my reasoning. Let $k$ be a field and $k \subseteq K$ a field extension. We say that $K$ is a ...
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0answers
27 views

Given a valuation on a field , does it always extend to a valuation on every extension field

Let $(G,+,\ge)$ be a totally ordered abelian group. Let $K\subseteq L$ be an extension of fields. Let $v : K\setminus \{0\} \to G$ be a valuation (https://en.wikipedia.org/wiki/Valuation_(algebra)) . ...
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0answers
34 views

Agebraically closed field equivalent definition

Is this a valid equivalent definition of an algebraically closed field? Let $ F $ be a field. Define a function $ f(x): F\longrightarrow F $ as a finite combination of the operations in $ F $ (...
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0answers
41 views

Affine Varieties over separably closed fields

Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$. On the ...
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1answer
17 views

Algebraically independent numbers and Archimedean field

What is the cardinality of the set of all algebraically independent numbers in $\mathbb{R}$? Can this be related to the total number of Archimedean fields possible as rational extensions of sets of ...
4
votes
2answers
57 views

What does it mean that field $\mathbb{F}_{p^n}$ “contains” the prime field $\mathbb{Z}_p$?

I have read in few books (example Computational Number Theory, page 77) that any extension field $\mathbb{F}_{p^n}$ "contains" as a subfield the prime field $\mathbb{Z}_p$? What exactly does "...
2
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2answers
35 views

Prove that if $p \not\equiv 1 \hspace{0.2 cm} (5)$ then $f(x) = x^{5} - 2$ has a unique solution in $\mathbb{F}_{p}$

To prove the statetament, i thought to define a linear application $$ \phi : \mathbb{F}_{p}^{*} \longmapsto \mathbb{F}_{p}^{*}$$ Define by : $f(x) = x^{5}$, studying the kernel of $\phi$ I noticed ...
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votes
1answer
37 views

Why does not generator of multiplicative group generate all the members of group? [duplicate]

Generator(g) of multiplicative group (Zp where p is primary) is an element the power of ...
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4answers
49 views

How to Show that a Field of Characteristic $0$ is Infinite [closed]

How can one prove that if a field $K$ has characteristic zero, then $K$ is infinite?
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1answer
25 views

A finite normal extension is also a splitting field

Let $E/K $ be a finite normal extension, then there exists $p(x)\in K[x] $ s.t. $E$ is the splitting field of $p(x) $. The proof goes as follows: $E=K(a_1,...,a_n) $ for some $a_1,...,a_n\in \Omega $ ...
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0answers
19 views

why separable algebra is a generalisation of separable field extension?

They are defined differently. Suppose we have a field $K$. We say a finite field extension $L$ is separable over $K$ iff the number of embeddings $L\hookrightarrow \bar K$ into the algebraic ...
2
votes
1answer
26 views

Example of a field of functions containing ln(x)

Up until now, I've mainly worked with the polynomial ring $\mathbb{R}[x_1,...,x_n]$ or the corresponding field of fractions. But I can't think of an example of a field of functions that contain non-...
2
votes
2answers
61 views

If a field $K$ (of characteristic 0) has no proper extensions of the form $K[\sqrt[n]{x}]$, is it algebraically closed?

Let $K$ be a field, assume characteristic 0 if this simplifies things. Suppose we know that for any positive integer $n$ and any $x \in K$ the polynomial $X^n - x$ has a root in $K$. Is $K$ ...
0
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2answers
60 views

Is there an efficient algorithm to find all zeros of systems of multivariate polynomial equations over a finite field?

I want my computer to solve large systems of multivariate polynomial equations over a finite field. The field is $\mathbb F_p$, where $p$ is a prime number. I heard that there is an algorithm using ...
2
votes
2answers
53 views

Is it 'rare' that $a$ and $a+1$ are conjugate (= have the same minimal polynomial)?

Let $a \in \bar{k}-k$, $k$ is a field of characteristic zero and $\bar{k}$ is an algebraic closure of $k$. Denote the minimal polynomial of $a$ by $m_a=m_a(t) \in k[t]$. Is it 'rare' that $m_a=m_{...
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votes
0answers
30 views

If $a \in L-k $ satisfies $k(a^n)=L$ (for all $n \geq 1$), then $L/k$ is Galois?

Let $k \subsetneq L$ be a finite separable field extension, and let $a \in L-k$ satisfy: For every $n \geq 1$, $k(a^n)=L$. In other words, all the non-zero powers of the primitive element $a$ are ...
4
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1answer
66 views

Galois group of $x^3-x^2-4$

In determining the Galois group of the polynomial $p(x) = x^3-x^2-4,$ I concluded that is must be the Klein-$4$ group as follows. First, $p(x) = (x-2)(x^2+x+2)$ and the roots of the irreducible ...
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0answers
51 views

$a,b \in \bar{k}$, such that $k(a)=k(b)=k(ab)$

Let $k$ be a field of characteristic zero, and let $a, b \in \bar{k}$ ($\bar{k}$ is an algebraic closure of $k$) be two distinct elements, such that $k(a)=k(b)$. Notice that $k(a)=k(b)$ implies that ...
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0answers
21 views

Normal extension of fields

I have to prove that $\mathbb{Q}(\sqrt{2},\sqrt{3}, u$) is a normal extension over $\mathbb{Q}$, where $u^2=(9-5\sqrt{3})(2-\sqrt{2})$. Could someone help me to prove this? I claimed that $\mathbb{Q}(\...
0
votes
2answers
49 views

Degree of extension of fixed field by infinite set of automorphisms.

If $G$ is a finite group of automorphism $E \rightarrow E$, then Dedekind-Artin theorem tells us that $[E:E^G]=\; \mid G \mid$ where $E^G$ is the subfield of $E$ fixed by the automorphisms of $G$. Is ...
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0answers
18 views

Handling collisions by redundancy - Newton’s identities

I am reading this paper (P2P Mixing and Unlinkable Bitcoin Transactions by Tim Ruffing, Pedro Moreno-Sanchez and Aniket Kate) and I need help in understand something in Section III-A, Handling ...
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1answer
22 views

Exercise about split closures (Galois Theory)

I am studing Galois Theory. I am using the books by Kaplansky, Fields and Rings. I am stuck doing this exercise: Let $M$ be a split closure of $L$ over $K$ ($M,L,K$ are all fields). Prove that $M=L_1\...
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1answer
59 views

Suppose that $F$ is a field with $27$ elements. Show that for every element $a \in F$, $5a = −a$

Suppose that $F$ is a field with $27$ elements. Show that for every element $a \in F$, $5a = −a$. I am not able to understand how to approach this problem.
1
vote
1answer
25 views

Minimal polynomial for normal closure

I came across this problem while studying Normal closures. Given $K=\mathbb{Q}$ and the polynomial $x^3-2\in K[x]$. $L/K$ is not normal where $L=\mathbb{Q}(2^{\frac13})$ since $\omega\not\in L $ ...
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0answers
43 views

Generator of $\mathbb{F}_{q}^*$ where $q$ is not prime.

I'm trying to find the generator of the multiplicative group $\mathbb{F}_{8}^*$, where $\mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would ...
4
votes
2answers
66 views

Why is $\mathbb{F}_{9}^*$ a multiplicative group

Since $\mathbb{F}_9$ is a field, its units $\mathbb{F}_{9}^* = (1,2,3,4,5,6,7,8)$ should form a multiplicative group. However in this group $3 \times 3 = 0 \notin \mathbb{F}_{9}^*$. I'm trying to ...
3
votes
0answers
44 views

About a problem of field extension in an algebra book by Fumiyuki Terada.

I am reading an algebra book by Fumiyuki Terada. There is the following problem in this book: $E_1, E_2, K$ are fields. $K$ is a subfield of $E_1$. $K$ is a subfield of $E_2$. $p, q$ are ...
3
votes
1answer
67 views

Show a polynomial is irreducible over field $F(\alpha)$

In the field $F(u)$, let $\alpha = \frac{u^3}{(u+1)}$. Consider the subfield $F(\alpha)$ of $F(u)$. Prove that the polynomial $f(x)=x^3-\alpha x-\alpha$ is irreducible over the field $F(\alpha)$. ...