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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Permuting roots in splitting fields

Currently, I've just started to study Field and Galois theory. In one of my textbooks, I have found the following (porbably important) theorem: If $K/F$ is a splitting field for the irreducible ...
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3answers
24 views

Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
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0answers
11 views

Quadratic Field of $Q[√−1]$…

Can someone show me a complex plane around the origin, with the points on the part of the complex plane which are quadratic integers in $Q[√−1]$. Another graph for $Q[√−3]$. And another for $Q[√−5]$. ...
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1answer
13 views

Primes in Quadratic Fields with Norm less than 6

What are the primes in $\mathbb Q[\sqrt{−1}]$ which have norm less than $6$? Also what primes in $\mathbb Q[\sqrt{−3}]$ have norm less than $6$, and the primes in $\mathbb Q[\sqrt{−5}]$? Which of them ...
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2answers
21 views

Image of archimedean place of a number field in $\mathbb C$

Let $L/K$ be a finite Galois extension of number fields and let $\phi$ be an embedding of $K$ into $\mathbb C$. Let $\psi_1$ and $\psi_2$ be two embeddings $L\to \mathbb C$ which extend $\phi$. ...
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3answers
41 views

Prove or disprove $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$

Want to prove or disprove this $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$ as communtative rings. I can show that $x^5-3$ and $x^5-9$ are irreducible in $\mathbb{Q}$, but I cannot go from ...
2
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1answer
17 views

Infiniteness of set of primes such $f$ have root $\mod p$ [duplicate]

Let $f \in \mathbb{Z}[x]$ be non constant. How to prove that exists infinitely many primes such $f$ have root in $\mathbb{Z/_{(p)}}$. I spent much time, but with no benefits.
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1answer
17 views

Is group of units of a polynomial ring only constant polynomial which is involved in R

Let R be a integral domain(or maybe field) edit : Let R be a field. The group of units of R[x] is $$ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 $$(or infinity) such ...
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1answer
13 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
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6answers
47 views

Prove that $x^3-2$ and $x^3-3$ are irreducible over $\Bbb{Q}(i)$

Let $F=\Bbb{Q}(i)$. Prove that $x^3-2$ and $x^3-3$ are irreducible over $F$. How do I go about this? Should I just say that the roots of $x^3-2$ are ...
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0answers
28 views

Confusion between polynomial in field and factorization.

Consider $f(x)=x^3+3x+2$ in $\mathbb{Z}_5[x]$ and we can see that this polynomial is irreducible over $\mathbb{Z}_5[x]$ since it has no zeros in $\mathbb{Z}_5$. After I read this example and found ...
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1answer
12 views

Real Closed Field, Degree of Monic/Irreducible

In a real closed field, $R$, why do all monic irreducible polynomials $f(x) \in R[x]$ have at most degree 1 or 2?
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1answer
52 views

Does $\alpha$ need to be transcendental over F?

In the book there is this exercise: Let E be an extension fiel of F, with $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over $F(\beta)$. Show that $\beta$ is ...
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4answers
44 views

Show that a polynomial over $\mathbb{Z}_{2}$ is irreducible

Given the polynomial: $p(x)=x^4+x^3+x^2+x+1$ over $\mathbb{Z}_{2}$, to show that it is irreducable, is it enough to show that $p(0)=p(1)=1$?
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1answer
19 views

How to prove subfield generated $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n) $

This is problem in Hungerford chapter 5: Fields and Galois Theory. Prove $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n)$ and $K[u_1,u_2,..u_{n-1},u_n]=K[u_1,u_2,..u_{n-1}][u_n] $ My ...
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2answers
36 views

What is $Q(x)$?

I do not really understand what $\mathbb{Q}(\pi)$ is here: Ofcourse we see that $\mathbb{Q}(\pi)$ is a field. But I have to "guesses" of what they mean, is one of them correct? 1. ...
1
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1answer
21 views

An alternate proof of the fact that a finite field extension is integral over the base field.

Let $F[x]/(p(x))$ be a field extension, where $F$ is a field, and $p(x)$ and irreducible polynomial in $F[x]$. We know that $F[x]/(p(x))$ is integral over $F$. The standard proof for this uses the ...
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0answers
32 views

Techniques to prove that two field extensions are distinct

I have been trying to create a family of pairwise distinct field extensions from a one-parameter family of irreducible polynomials, but have no idea how to prove that they are distinct. One pair is $f ...
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4answers
68 views

Are these two fields the same?

I wanted to know if the field $\mathbb{Q}(i\sqrt{7}) = \mathbb{Q}(\sqrt{7}, i)$ are the same. I don't think they are because $i \notin \mathbb{Q}(i\sqrt{7})$?
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1answer
20 views

Proving something is not a Normal Extension

Let $M = \mathbb{Q}(\sqrt{3}, i\sqrt[4]{5})$ be an extension of $\mathbb{Q}$. Then work out the basis of $M$ over $\mathbb{Q}$ and show that the extension $M/\mathbb{Q}$ is not a normal extension. So ...
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0answers
20 views

compute the grades over $\mathbb{Q}$ [duplicate]

Let $p_{1}$ $\neq$ $p_{2}$ $\neq$ $p_{3}$ prime numbers. Compute the grades over $\mathbb{Q}$ of the extension fields $\mathbb{Q} ( \sqrt{p_{1}}, \sqrt{p_{2}})$ and $\mathbb{Q} ( \sqrt{p_{2}}, ...
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1answer
56 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all $2 × 2$ matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? We know, in the ring $\mathbb{Z}$, ...
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0answers
42 views

Whether the number $a$ is algebraic over $\mathbb{Q}$ [on hold]

Is the number $a = \displaystyle \left( \frac{(1 + \sqrt[3]{7})^{\tfrac{7}{5}}}{(\sqrt[3]{7} - 7)^3 + 77} \right)^{13}$ algebraic? If so, is algebraic degree of $a$ bounded by $15$?
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votes
1answer
12 views

Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Let $F$ be a field, and $p(x)\in F[x]$ be an irreducible polynomial. Suppose $\alpha$ is a root of $p(x)$. Show that if $q(x)$ is a polynomial such that $\deg q(x) < \deg p(x)$ and $q(\alpha)=0$, ...
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votes
2answers
24 views

Relation between reduced finite algebra, prime ideal and field extension

Is it true that if $L$ is a reduced finite dimensional commutative algebra over a field $K$ (which is finite or of characteristic $0$) and if $\mathfrak p$ is a prime ideal of $L$, then ...
0
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4answers
66 views

Characteristic of a Finite Integral Domain

I am a little confused as how to approach this problem. The title of this problem is the title of the section which it comes from. However, there is no information that the given integral domain is ...
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1answer
16 views

I need to show that if $K$ is of characteristic $0$,the algebra $A$ has a primitive generator.

Let $K$ be a field and $A$ a reduced K-algebra of finite dimension over $K$. I need to show that if $K$ is of characteristic $0$, $A$ has a primitive generator (i.e. $A=K[x], x \in A$) I've proved ...
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2answers
24 views

Why is the fixed field of this automorphism is $Q(\pi^2)$?

Let $\sigma:Q(\pi)\rightarrow Q(\pi)$ be an auorphism fixing $Q$ such that $\sigma(\pi)=-\pi$. Let $F$ be the fixed field. Then it is obvious that $Q(\pi^2)\subset F$. However, why Why is ...
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2answers
74 views

How do I show that $\sqrt{5}\notin \mathbb{Q}(\sqrt{2},\sqrt{3})$?

How do I show that $\sqrt{5}\notin \mathbb{Q}(\sqrt{2},\sqrt{3})$? Since $X^2-5$ is the minimal polynomial of $\sqrt{5}$ over $\mathbb{Q}$ and its degree is not relatively prime to ...
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2answers
36 views

$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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0answers
24 views

Proof of primitive element theorem for $F$ finite

I am trying to understand the proof of primitive element theorem, in particular this statement: Let $E/F$ be a field extension such that there are finitely intermediate fields containing $F$. Then $E ...
2
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1answer
36 views

Algebraic subfield of transcendental extension

I was recently thinking about whether it is possible to generate an infinite dimensional algebraic extension over a base field using just finitely many transcendental elements. Specifically, given a ...
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1answer
29 views

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials in $\mathbb Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in ...
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votes
1answer
139 views

Is this element algebraic over $\mathbb{Q}$?

I'm trying to see whether the following element is algebraic over $\mathbb{Q}$ and if it is - try and find its degree: $$ a = \left(\frac{(1 + \sqrt[3]{7})^{7/5}}{(\sqrt[3]{7} - 7)^3 + ...
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0answers
16 views

Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
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1answer
27 views

Construct the Normal Closure for this extension

I am trying to find the normal closure for the following extension $\Bbb Q\subset\Bbb Q(t)$, where "$t$" is a zero of $x^3-3x^2+3$ and $\Bbb Q$ are the rational numbers. I know the normal closure ...
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0answers
29 views

Degree $5$ Irreducible Polynomial Solvable by Radicals and Abelian Extension

Consider an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is an ...
2
votes
1answer
34 views

Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
2
votes
1answer
53 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, but $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not

How can I show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ but that $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not? I am kind of lost with Galois Theory. Thanks
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0answers
30 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
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3answers
31 views

Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
2
votes
2answers
46 views

Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

I do not understand why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$. I know that $x^{4}+1$ is irreducible $(f(x + 1) = (x + 1)^4 + 1$ is Eisenstein at $2$) and it has the roots: ...
1
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1answer
29 views

Galois group of $f$ is cyclic if $\deg f$ is prime

Hello I am learning Galois Theory by myself and got lost in the following exercise: Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a ...
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0answers
24 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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1answer
37 views

Squares in a field with $q^n$ elements.

Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
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2answers
30 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
3
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0answers
56 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
2
votes
1answer
29 views

$a$ and $1+a^{-1}$ have same degree over $F$ if $a$ is algebraic over $F$

Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$. Not really sure where to start for this one. I know that I have to show that ...
0
votes
1answer
12 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
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2answers
36 views

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$.

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$. Let $f(x)=x^2+x+1$. Then $f(a)=a^2+a+1=0$. To show equality of the two fields, we need ...