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
45 views

Determine the degree of the extension over Q

Determine the degree of the extension $Q(\sqrt{3+2 \sqrt{2}})$ over Q. I can see that $$3+2 \sqrt{2} = (1+ \sqrt2)(1+ \sqrt2) =(1+ \sqrt2)^2$$ does that mean $$x^2 -(1+ \sqrt2)^2)$$ has a degree $2$. ...
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1answer
11 views

Determine how many elements in $F_2[x]/(g(x))$, $F_2[x]/(h(x))$, $F_3[x]/(g(x))$ and $F_3[x]/(h(x))$

let $g(x) = x^2+x-1$ and let $h(x) = x^3-x+1$ obtain fields $4$, $8$, $9$, and $27$ elements by adjoining a root of $f(x)$ to the field $F$ where $f(x)=g(x)$ or $h(x)$ and $F = F_2$ or $F_3$. The ...
1
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1answer
18 views

Automorphism of $\mathbb{Q}({\zeta_n})/\mathbb{Q}$

I came across the theorem where, for $n=p^{a_1}\cdots p^{a_m}$: $Gal(\mathbb{Q}({\zeta_n})/\mathbb{Q})\simeq$ $Gal(\mathbb{Q}({\zeta_{p^{a_1}}})/\mathbb{Q})\times ...
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1answer
62 views

Find Primitive Root for Polynomial Field

Can someone help me get started on the problem below: Recall that $\mathbf{F}_{p^k}$ can be realized as $\mathbf{F}_p[x]/P(x) \cdot \mathbf{F}_p[x]$ where $P(x)$ is a polynomial of degree $k$ with ...
2
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1answer
44 views

dimension of quotient by algebraically independent elements

Let $f_1,\dots,f_s$ be algebraically independent polynomials of $A:=k[x_1,\dots,x_n]$, $s \le n$. Recall that algebraically independent means that there is no non-zero polynomial $g \in ...
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1answer
49 views

What does $k^*/k^{*^2}$ mean?

I'm trying to get a more concrete understanding of what these elements 'look like.' Here $k$ is a field, $k^*$ is multiplicative group, and $(k^*)^2$ consists of the squares in $k^*$.
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0answers
11 views

Normality of towers of fields

If I have a tower of fields $F \subseteq E \subseteq K$ and K is finite and normal over F. I know that E needn't be normal over F because for example $\mathbb{Q}(\zeta_6, \sqrt{2})$ is normal over ...
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1answer
22 views

what i want to know is how to compute the powers of $\theta$ in $F_2$ and also how many powers am i looking to compute. How can i find such powers [duplicate]

This a new chapter that we are learning and the teacher is flying through it and this are also new concept that i have just learn and i was wondering if i can have some guidance in this problem. ...
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2answers
23 views

Show that the field of real numbers has an infinite proper subfield but no finite subfields.

Show that the field of real numbers has an infinite proper subfield but no finite subfields. $\mathbb{Q}$ is an infinite subfield and as $|\mathbb{Q}| < |\mathbb{R}|$, it is also a proper subfield. ...
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1answer
16 views

Isomorphisms in characterisation of Galois extension

My definition of an extension $M/K$ to be Galois is that $Gal(M/K)$ only fixes things in K. I'm trying to prove that this is equivalent to $M/K$ being normal and separable. I know that fact that if ...
4
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2answers
134 views

Using Zorn's lemma to show that every field has an algebraic closure.

You may have seen that I posted this proof with some questions earlier today. But I found the answer to most of them. Now I have just one question regarding this proof, so I thought it would be better ...
0
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1answer
26 views

Field extension and automorphism proof understanding.

Suppose $F\subseteq L $ is any field extension, $f(x) \in F[x]$, and $b_1,b_2,...b_r$ are distinct roots of $f(x)$ in L. Prove the following statements: 1) If $\sigma$ is an automorphism of L that ...
2
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1answer
17 views

Infinite number of intermediate fields between K(u,v) and K

$K$ is an infinite field with char $K =p >0$ and suppose $L=K(u,v)$ where $u^p, v^p\in K $and $[L:K]=p^2$. Show that there exist infinite number of distinct intermediate fields between $K$ and $ ...
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1answer
26 views

Purely inseparable subextension of $F(x)$

Suppose that $F$ is a field of characteristic $p>0$. Prove field extension $F(x^p)\subset F(x)$ is purely inseparable. I think we should first prove that is inseparable, and then show it is ...
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0answers
36 views

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ [duplicate]

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ where $p_1,p_2,\cdots ,p_k,p$ are distinct primes. My try: For $k=1, ...
5
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1answer
38 views

$x^p -x-c$ is irreducible over a field of characteristic p if it has no root in the field

Let $c$ be an element of a field $F$ of characteristic $p$ (prime). Then how to show that $x^p -x-c$ is irreducible over $F$ if it has no root in $F$. I was trying using contradiction and by ...
4
votes
1answer
42 views

Galois Extension whose Galois Group is $\mathbb{Z}_2\oplus\mathbb{Z}_4$

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Let $E/F$ be a Galois extension with Galois group isomorphic to ...
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2answers
33 views

“Implicit” condition about separability of a quartic polynomial

Here is an exercise in Hungerford's Algebra, page 277 Ex.12. Let $K$ be a subfield of real numbers and $f \in K[x]$ an irreducible quartic polynomial(of degree 4). If $f$ has exactly two roots, the ...
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1answer
67 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
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1answer
37 views

Minimal polynomial of $\sqrt[3]{2} + \omega$ over $\mathbb{Q}.$

Is the polynomial $f(x) = x^9 - 9x^6 - 27x^3 - 27$ irreducible over $\mathbb{Q}?$ I think it is because of Eisenstein's applied to the prime $3.$ Is it the minimal polynomial of $x = e^{2 \pi i/3} + ...
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2answers
79 views

Why is F($\beta$) a subfield of F($\alpha$)?

There is a corollary in my book that says: If E is an extension field of F, $\alpha \in E$ is algebraic over $F$, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ divides $\deg(\alpha,F)$. In ...
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1answer
39 views

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$ I simplified $x^{100} + 2x + 10$ to $x^{15} + 2x + 10$ and $x − 11$ to $x+6$ to be in $\mathbb Z_{17}$. ...
3
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0answers
34 views

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ...
3
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2answers
69 views

Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel

I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group ...
0
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1answer
23 views

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite. $P$ is a maximal ideal of $R/P$ is a field. I ...
2
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1answer
66 views

Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
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2answers
31 views

Factor Ring question and finding maximal ideals of $\mathbb{Z}\times\mathbb{Z}$

What is the maximal ideal of $\mathbb{Z}\times\mathbb{Z}$? I think since $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, it seems like that ...
2
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2answers
34 views

Maximal ideals and prime ideals of $\mathbb{Z}/2 \times \mathbb{Z}/2$?

I think there are 3 ideal and maximal primes. $<(0,1)>$ since factor group over $<(0,1)>$ is isomorphic to $\mathbb{Z}/2$, which is field and integral domain. And same reason for ...
2
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3answers
46 views

$\mathbb Q [\sqrt{2} i]$ contains neither $\sqrt[4]{2}$ nor $\sqrt{2}$

I want to prove that $x^4-2$ is irreducible over $\mathbb Q [\sqrt{2} i]$. In order to verify it has no linear factors and quadratic factors, I need to show $\mathbb Q [\sqrt{2} i]$ contains neither ...
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2answers
60 views

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 (probably important) theorem: If $K/F$ is a splitting field for the irreducible ...
1
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3answers
31 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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1answer
22 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 ...
1
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2answers
26 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 $\phi$. ...
3
votes
3answers
54 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
votes
1answer
22 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
19 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
25 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
60 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
29 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
13 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
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1answer
58 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
48 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
23 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
43 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
22 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
1answer
53 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
76 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
21 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
60 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 $\mathbb{Z}$, ...