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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0
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1answer
37 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
0
votes
1answer
39 views

Prove that factor rings are fields [closed]

Prove that factor rings $\mathbb{Z}_3[x]/(x^3 + x^2 +2)$ and $\mathbb{Z}_3[x]/(x^3 -x +1)$ are fields, and these fields are isomorphic.
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0answers
19 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
-1
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2answers
76 views

$\mathbb Q(\sqrt2) \not \cong \mathbb Q(\sqrt[3]{2})$ [closed]

Prove that all fields $\mathbb Q(\sqrt2)$ and $\mathbb Q(\sqrt[3]{2})$ are not isomorphic
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vote
1answer
21 views

What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
1
vote
1answer
33 views

Intermediate fields of $X^p - 2 $

I've been working on an exercise I have to do for my algebra course. Exercise: Let p be prime and $L$ the splitting field of $ f = X^p - 2$ over $\mathbb{Q}$. a) Show that $ Gal(L/\mathbb{Q})$ is ...
0
votes
2answers
56 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
2
votes
3answers
103 views

Field extensions with(out) a common extension

Let $K$ be a field having two field extensions $L\supseteq K$ and $M\supseteq K$. Does there exist a field $N$ along with embeddings $L\to N$ and $M\to N$, such that the diagram $$ \require{AMScd} ...
3
votes
1answer
66 views

Find all the fields between $\mathbb{Q}$ and the splitting field of $x^4 + 81$

Let $f(x)=x^4+81 \in \mathbb{Q}[x]$. Find the splitting field $E$ of $f(x)$ and the extension degree $[E:\mathbb{Q}]$. Find all the fields $L$ with $\mathbb{Q} \leq L \leq E$. Are the roots of ...
2
votes
0answers
72 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...
11
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2answers
182 views

Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$.

Is $\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$? Say $L=\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)$ and $K=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$. It is easy to show that ...
1
vote
2answers
69 views

Degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$

Question is to compute the degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$. We have ...
0
votes
0answers
9 views

Explanation of why a differential field extension must be trancendental

I'm attempting to follow the steps of this proof that $e^{x^2}$ has no antiderivative and there is one step that I'm not quite understanding. They state: If K is a differential field then $K_C=\{r ...
0
votes
1answer
22 views

Find extension degree

Let $\zeta =e^{\pi i/12}$. Find the extension degree of $\mathbb{Q}\leq \mathbb{Q}(\zeta)$ Show that $\mathbb{Q}(\zeta)=\mathbb{Q}(\sqrt{2} , \sqrt{3} , i)$ $\zeta$ is a root of $x^{24}-1$ ...
0
votes
1answer
42 views

Show that it is an element of $L$

Let $L$ the subfield of the complex with $\mathbb{Q}\leq L$ a normal extension. If $a=\sqrt{5}-\sqrt[3]{2}\in L$, show that $\omega=e^{2\pi i/3}\in L$. I have done the following: ...
1
vote
0answers
32 views

The equation has a solution iff $(k,q-1)=1$

Let $p^n$, where $p$ is a prime, and $k \in \mathbb{N}$. Let $a$ the generator of the cyclic group $\mathbb{F}_p^{\star}$, where $\mathbb{F}_q$ the finite field with $q=p^n$ elements. Show that the ...
2
votes
0answers
38 views

If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
0
votes
2answers
76 views

What is the splitting field of $t^4+2$?

I currently beginning the study if Galois theory but my understanding of the construction of splitting fields could be better. So I must ask if I could see the steps in constructing the splitting ...
0
votes
1answer
27 views

The polynomial splits into distinct factors in $F[x]$

Let $F$ be a field of characteristic $p$. Show that if $x^p-x-a$ is reducible in $F[x]$, then it splits into distinct factors in $F[x]$. I have done the following: We want to show that ...
0
votes
1answer
49 views

$X$ is algebraic over $E$

Let $X$ be transcendental over a field $F$, and let $E$ be a subfield of $F(X)$ properly containing $F$. Prove that $X$ is algebraic over $E$. Could we maybe use also the following?? Let $f(x) \in ...
1
vote
2answers
28 views

$G$ is isomorphic to $S_3$

Show that the Galois group of the splitting field $F$ of $X^3-7$ over $\mathbb{Q}$ is isomorphic to $S_3$. I have found that the the Galois group is the following: $$G=\{\tau_{ij}, i=1,2,3, ...
1
vote
1answer
25 views

Finite fields and order [duplicate]

If $F$ is a finite field then is it necessary that $|F|=p^n$ for some prime $p$ and positive integer $n$? I know that given prime $p$ and positive integer $n$, there is a field such that $|F|=p^n$? ...
0
votes
1answer
51 views

Show that the intersection is $F$

Let $F$ be a field of characteristic $0$. Show that $F(x^2) \cap F(x^2-x)=F$. Could you give me some hints how I could do that??
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0answers
17 views

Normal soluble field towers

(All fields below are subfields of the field $\mathbb{C}$). I am looking for a proof (or disproof) of the following statement: Let $L:K$ be a normal extension and $K = N_{0} \subset N_{1} \subset ...
4
votes
2answers
42 views

$\{p_i\}$ generate the $k$-algebra of symmetric polynomials in $k[t_1, \dots, t_n]$ and are algebraically independent over $k$

Let $k$ be a field of characteristic $0$. For $j \ge 0$, let $p_j = t_1^j + \dots + t_n^j \in k[t_1, \dots, t_n]$. Prove that $p_1, \dots, p_n$ generate the $k$-algebra of symmetric polynomials in ...
2
votes
1answer
49 views

Field Extensions and Number of Isomorphisms

The picture above is from Dummit and Foote, Third Edition. Later in the book, we find Clearly, the condition of equality is not necessary, as seen by taking the polynomial $ f(x) = ( x ^{2} ...
0
votes
3answers
59 views

Show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$ [closed]

How could we show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$ ?? Could you give me some hints??
1
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1answer
40 views

Isomorphism of Extensions

Let $\mathbb{L}$ an extension field of $\mathbb{K}$ and $\alpha, \beta\in\mathbb{L}$. If they have the same minimal polynomial than $\mathbb{K}(\alpha)\simeq\mathbb{K}(\beta)$, because if: ...
1
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0answers
24 views

Orbits of elements of the field under the action of a subgroup of field automorphisms

Let $G$ be a subgroup of the group of automorphisms of field $E$, $\mid G\mid = n$. Can we find such $\alpha\in E$ that its orbit under $G$ consists of $n$ different elements? If $E$ is a Galois ...
1
vote
1answer
60 views

Show the quotient ring R/I is not a field

Studying for an exam in Algebra. Let $R=\mathbb{Z}[i]$ with the usual normfuction $N, z = 5+3i$ and $I = \, <z>$ Show that z isn't a prime element in $R$ and that $R/I$ isn't a field. I ...
5
votes
1answer
78 views

Algebraic closure of the rational inside a quotient of product of finite fields

I'm trying to solve the following exercise: " Consider the ring $R = \prod_{p} \mathbb{F}_p$, where $p$ runs over all prime numbers and $\mathbb{F}_p$ is a field with $p$ elements. Show that there ...
1
vote
1answer
37 views

Specific question on imaginary quadratic field [closed]

How to solve the following question?! Let $K$ be an imaginary, quadratic field and let $L/K$ be a Galois extension. If $\tau$ is complex conjugation, show that: (a) $L/\Bbb Q$ is Galois iff ...
0
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2answers
59 views

Why can't this happen in fields $K$ with $\operatorname{char}(K)=0$?

Let $f(x) \in K[x]$ irreducible. Then: $f(x)$ separable $\Leftrightarrow $ $(f, f')=1$ $f$ is irreducible so $$ (f,f')=\left\{\begin{matrix} f\\ 1 \end{matrix}\right. $$ $f(x)=a_0+a_1 x + \dots + ...
1
vote
1answer
40 views

Must algebraic extensions of the same degree have subfields of the same degree?

Let $\mathbb F$ be a field and let $\mathbb K_1$ and $\mathbb K_2$ be finite extensions of $\mathbb F$ with the same degree, that is, $[\mathbb K_1:\mathbb F]=[\mathbb K_2:\mathbb F]$. Now, assume ...
1
vote
2answers
31 views

Polynomial for each extension degree

Let $E$ the splitting field of a polynomial in $\mathbb{Q}[x]$ of degree $3$, then $[E:\mathbb{Q}]=1,2,3,6$. I am asked to give an example for each case... Are the following correct?? ...
0
votes
0answers
43 views

Question on two isomorphic simple algebraic extension

Let F be a field. Suppose that $\alpha$ and $\beta$ are two algebraic elements over F such that $F(\alpha)$ is isomorphic to $F(\beta)$. What is the relation between $\alpha$ and $\beta$?
3
votes
0answers
43 views

Finite Ring with unity and no zero divisors is field

I would like to know if someone can help me with this. "Show that a finite ring $R$ with unity $1\neq 0$ and no divisors of 0 is a field." The original exercise asked me to show that it was a ...
1
vote
0answers
27 views

Field of polynomials mod n?

I have a few questions and i am looking for some clarification. 1) Is it correct that one can define a field $(Z_n, +, X)$ of integers mod $n$, where all the elements are integers $a$ such that ...
2
votes
1answer
38 views

Find the cardinality of $\mathbb{F}_2$ adjoin a root of $X^4 + X + 1$

Consider the irreducible polynomial $g = X^4 + X + 1$ over $\mathbb{F}_2$ and let $E$ be the extension of $\mathbb{F}_2 =\{0,1\}$ with a root $\alpha$ of $g$. How many elements does $E$ have? ...
0
votes
3answers
86 views

Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
6
votes
1answer
45 views

Is there a faster way to factor $X^{12}-1$ over $\mathbb{F}_5[X]$?

Problem: Factor $X^{12}-1$ into irreducibles in $\mathbb{F}_5[X]$. This problem appeared on a past qual and took me awhile to do. While I solved it, I'll need to be able to do problems like this a ...
1
vote
0answers
45 views

How to show that $[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9$

How can I show that $$[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9?$$ My idea is: $\sqrt[3]2$ has for its minimal polynpmial $X^3-2$ over $\mathbb Q(i\sqrt 3)$, which I justify by: ...
5
votes
6answers
494 views

Where do the coefficients belong to?

We have the polynomial $f(x)=x^3+6x-14 \in \mathbb{Q}[x]$. We have that $f(x)$ has exactly one positive real root $a$. That means that $f(x)$ can be written as followed: $$f(x)=(x-a)(x^2+px+q)$$ ...
1
vote
3answers
64 views

annihilator polynomial of a multiplicative group in a Field?

Consider the annihilator polynomial of a multiplicative group $H$ of a field $\mathbf{F}_q$. $$A(x) = \prod_{\alpha\in H} (x-\alpha)$$ I read somewhere that this polynomial can be written as $A(x) ...
0
votes
0answers
97 views

change the matrix when we extend the field

Let $M$ be an $F_pC_q$- module represented by the matrix $$\left( \begin{matrix} a & b\\ c & d \end{matrix}\right)$$ i.e., $m_1 g=am_1 + bm_2$ and $m_2g=cm_1 + dm_2$ where g is the ...
1
vote
1answer
29 views

Two different matrix representations of complex numbers

There are two different ways to represent a complex number with $2 \times 2$ real matrices: $$ \rho: \mathbb{C} \rightarrow M_2(\mathbb{R}) \qquad \rho(z)=\rho(a+ib)= \left[ \begin{array}{ccccc} ...
2
votes
0answers
51 views

A book for advanced field theory

I am searching for an alternative text to chapter 5 of Bourbaki for field theory, that covers, for example, separable and inseparable degrees. I know the basics about field theory and Galois theory. ...
0
votes
0answers
32 views

Solution of the equation $x^r=a$

Let $F_{p^n}$ be the field with $p^n$ elements. Suppose $p^n-1=q_1^{a_1}...q_k^{a_k}$ where $q_i$ are distinct primes. Find the no. of integers $r\in\{0,1,...,p^n-2\}$ for which the equation $x^r=a$ ...
0
votes
1answer
38 views

Dual basis in a finite separable extension

I am reading the book Algebras, Rings and Modules, volume 1, by M. Hazewinkel and at the page 193 there is a proof about why the integral closure of a ring in a separable finite extension L over $k$ ...
1
vote
2answers
41 views

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.($F_5$ is the field with $5$ elements.) My Work: Since $(0,1)$ does not have an inverse, $F_5\times F_5$ is not a ...