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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2
votes
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.
1
vote
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 ...
0
votes
1answer
12 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 ...
6
votes
6answers
46 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 ...
0
votes
0answers
27 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 ...
-1
votes
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?
3
votes
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 ...
0
votes
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 $p(0)=p(1)=1$?
1
vote
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 ...
1
vote
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
vote
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 ...
2
votes
0answers
29 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 ...
4
votes
4answers
66 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 $i \notin \mathbb{Q}(i\sqrt{7})$?
3
votes
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 ...
0
votes
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}}, ...
2
votes
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 $\mathbb{Z}$, ...
-2
votes
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 $15$?
0
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 $q(\alpha)=0$, ...
0
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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: ...
0
votes
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
votes
1answer
35 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 ...
0
votes
1answer
24 views

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

How do I prove that $X^{p^n}-X$ is the product of all monic prime polynomials in $Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in \bar ...
0
votes
1answer
134 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 + ...
1
vote
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 $\implies$ ...
0
votes
1answer
26 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 ...
0
votes
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
1
vote
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 ...
0
votes
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
vote
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 ...
2
votes
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 ...
0
votes
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 $r^m=1.$
3
votes
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
votes
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?
0
votes
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 ...
1
vote
2answers
52 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
1
vote
2answers
14 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
1
vote
2answers
28 views

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$?

How do I find all conjugates of $\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$? I have shown that $\sqrt{1+\sqrt{2}}$ is a root of $X^4 - 2X^2 - 1$ and this polynomial is irreduciable in $\mathbb{Q}[X]$, ...
2
votes
2answers
40 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
2
votes
1answer
68 views

Showing that $40^{\circ}$ is not constructible

Show that $40^{\circ}$ is not constructible. Attempt We note that $\cos 120^{\circ}=-\frac{1}{2}$ and that it also equals $4\cos^340^{\circ}-3\cos40^{\circ}$, which is obtained by using the ...
1
vote
0answers
19 views

What would be a value of $X$ under an automorphism of $F(X)$ over $F$?

Let $\sigma:F(X)\rightarrow F(X)$ be a field automorphism fixing $F$. What would be an value of $\sigma(X)$? Since $X$ is transcendental over $F$, $\sigma(X)$ is a transcendental over $F$ and ...