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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157 views
Irreducible polynomials of real algebraic numbers
Suppose $\alpha$ is a real algebraic number with the property that its irreducible polynomial over $\mathbb{Q}$ is not a binomial, i.e., it is not of the form $x^n-q$ for some $n\geq 1$ and ...
6
votes
1answer
106 views
generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$
I would like to find a generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$. This is a field since $x^{2}+3x+3$ is irreducible, so every coset with $bx+a\not=0$ as a ...
6
votes
1answer
113 views
An “independence” condition on two algebraic elements over $K$.
Let $K$ be a field and let $a,b\in \overline K$ be algebraic elements.
I've stumbled upon a certain condition on $a,b$, which I feel could be considered an "independence" condition. I would like to ...
6
votes
1answer
314 views
$K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable
I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal ...
6
votes
2answers
421 views
How to find irreducible polynomials over $\mathbb{Q}(i)$ with prescribed Galois group?
Here is my recent homework question:
For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example ...
6
votes
1answer
123 views
Calculating Separable Closures
In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion ...
6
votes
1answer
118 views
For which $k$ do the $k$th powers of the roots of a polynomial give a basis for a number field?
Let $f \in \mathbb{Q}[x]$ of degreee $d$ be irreducible, with roots $\alpha_1,\ldots, \alpha_d$. One particular basis for the field extension of $\mathbb{Q}$ obtained by adjoining the roots of $f$ is ...
6
votes
1answer
44 views
Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is ...
6
votes
1answer
149 views
Calculating the norm of an element in a field extension.
Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element ...
6
votes
0answers
418 views
Galois closure of a $p$-extension is also a $p$-extension
I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads:
Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose
Galois group is a ...
5
votes
3answers
319 views
Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$?
Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$ when $\mathbb{Q}(\alpha)$ is a $p$th degree Galois extension of $\mathbb{Q}$?
($p$ is prime)
I got stuck with this ...
5
votes
3answers
260 views
Field with natural numbers
To make sure that we are talking about the same, I would like to post the relevant definitions I know first.
Definitions:
A pair $(G, +)$ where $G$ is a set and
$+: G \times G \rightarrow G$
is ...
5
votes
3answers
310 views
Isomorphisms: preserve structure, operation, or order?
Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two ...
5
votes
4answers
193 views
Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
By Gauss' Lemma, $p(x)$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in $\mathbb Z[x]$. We can look at ...
5
votes
2answers
245 views
How do I prove that this polynomial is irreducible?
How do I prove that $x^4+1$ is an irreducible polynomial over $\mathbb Q$? I've already tried the Eisenstein criterion which gives to me any results to solve this question, I need help here.
Thanks
5
votes
2answers
175 views
How many fields inside $\mathbb R$?
i.e. what is cardinality of $\{A \mid \ A\subset \mathbb R, A \text{ is a field} \}$?
5
votes
2answers
463 views
Tensor product and compositum of fields
Let E/k, F/k be two arbitrary field extensions of k. My question is:
Is there a field extension M/k s.t. E/k, F/k are subextensions of M/k? Alternatively, can we talk about compositum fields without ...
5
votes
4answers
255 views
Minimal polynomial of the root of algebraic number
I have obtained the minimal polynomial of $9-4\sqrt{2}$ over $\mathbb{Q}$ by algebraic operations:
$$ (x-9)^2-32 = x^2-18x+49.$$
I wonder how to calculate the minimal polynomial of ...
5
votes
3answers
399 views
If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]
Possible Duplicate:
Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field
I need to prove this result, but the only starting point I think of is to ...
5
votes
2answers
146 views
In an ordered field, must 1 be positive?
In an ordered field, must the multiplicative identity be positive? Or must it be defined as such?
5
votes
1answer
223 views
Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?
Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ?
My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism...
What is the 'standard' ...
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votes
2answers
174 views
If $X^n+Y^n+1$ is reducible, is the degree divisible by the characteristic?
I was playing around with the Frobenius map, and made a small observation.
Suppose $F$ is a field, and $F[X,Y]$ is the corresponding polynomial ring in two indeterminates. If $\text{char}(F)=p$ ...
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votes
3answers
367 views
Algebraic Closure of Puiseux Series
Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
5
votes
6answers
134 views
Show that $x^4 +1$ is reducible over $\mathbb{Z}_{11}[x]$ and splits over $\mathbb{Z}_{17}[x]$.
Reduction into linear factors $\mathbb{Z}_{17}[x]$:
This part is not too hard: $x^4 \equiv -1$ mod 17 has solutions: 2, 8, 9, 15 so
$(x-2)(x-8)(x-9)(x-15) = x^4 -34 x^3 +391 x^2-1734 x+2160 \equiv ...
5
votes
3answers
645 views
On a formula of the norm of an element of a finite extension of a field
Theorem
Let $F$ be a field.
Let $K$ be a finite extension of $F$.
Let $[K : F]_i$ be the inseparable degree of $K/F$.
Let $\bar{K}$ be an algebraic closure of $K$.
Let $S$ be the set of $F$-embeddings ...
5
votes
6answers
244 views
Abstract algebra book recommendations for beginners.
I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
5
votes
3answers
458 views
Irreducibility and Splitting Fields
Show that over any field $F$, the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors.
Edited:
This is my attempt: Let $f(x)=x^3-3x+1$. Let $a_1,a_2,a_3$ be the roots of ...
5
votes
3answers
147 views
Count of elements in $\Bbb{Z}_7[x]/(3x^2+2x)$
Hi I have some problem how to get count of elements in $\Bbb{Z}_7[x]/(3x^2+2x)$. I think there belong to only polynomials which are indivisible with $3x^2+2x$ ($\gcd=1$). I think it is so as far I ...
5
votes
1answer
321 views
What is a maximal abelian extension of a number field and what does its Galois group look like?
How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$?
I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
5
votes
1answer
153 views
For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$
Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that differs from $F$. Why is $x$ algebraic over $K$?
Thanks a lot!
5
votes
2answers
306 views
Minimal polynomial of $1 + 2^{1/3} + 4^{1/3}$
I am attempting to compute the minimal polynomial of $1 + 2^{1/3} + 4^{1/3}$ over $\mathbb Q$. So far, my reasoning is as follows:
The Galois conjugates of $2^{1/3}$ are $2^{1/3} e^{2\pi i/3}$ and ...
5
votes
2answers
157 views
How does extending a field affect matrix similitude?
Suppose we have fields $\mathbb{K}_1, \mathbb{K}_2$ s.t. $\mathbb{K}_1 \subset \mathbb{K}_2$. On a paper I'm reading there is a flashy reference to some algebraic results concerning similitude of ...
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votes
2answers
377 views
Trace as Bilinear form on a field extension
Can anyone help with this:
If $L/K$ is a finite field extension, and we have a $K$-bilinear form given by $$(x,y)\mapsto Tr_{L/K}(xy)$$ then the form is either non-degenerate or $Tr_{L/K}(x)=0$ for ...
5
votes
1answer
479 views
Problem in Jacobson's Basic Algebra (Vol. I)
It is section $4.4$, exercise number $5$. It says the following: Let $F$ be a field of characteristic $p$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if ...
5
votes
2answers
827 views
reducible polynomial modulo every prime
how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$.
For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
5
votes
1answer
358 views
Frobenius homomorphism
Its easy to proof that any non-zero field homomorphism is injective:
Proof
Assume that $\exists a, b\in F: a\neq b~~and~~\psi(a)=\psi(b)$ then:
$$\psi(1)=\psi((a-b)^{-1}(a-b))=\psi((a-b)^{-1})\cdot ...
5
votes
2answers
162 views
Is $\mathbb{R}(X+Y)\subseteq\mathbb{R}(X,Y)$ a purely transcendental extension?
Is there a nice, short and elementary argument that the field extension $\mathbb{R}(X+Y)\subseteq\mathbb{R}(X,Y)$ is purely transcendental?
Obviously, $\mbox{tr ...
5
votes
4answers
96 views
$F:= \{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \}$ closed under addition, subtraction, multiplication, and division
I am in my math class and I came across this problem on my past midterm. How can we prove that $F:=\{ a+b\sqrt{7} \mid a,b \in \mathbb{Q} \} $ is closed under addition, subtraction, multiplication, ...
5
votes
3answers
189 views
Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible
This is an exercise for the book Abstract Algebra by Dummit and Foote
(pg. 519): show $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible for $a\neq-1,0,2$
I need help with this exercise, I don't have an ...
5
votes
1answer
109 views
Is $i\notin \mathbb{Q}(\zeta_p)$ for all odd primes $p$?
My main question is the title: for an odd prime $p$, denote a primitive $p^{\text{th}}$ root of unity by $\zeta_p$. Is it true that $i$ is not contained in the cyclotomic extension ...
5
votes
1answer
198 views
A characterization of finite purely inseparable extensions of fields
Let $K/k$ be a finite extension of fields. Let $A=K \otimes_k K$.
An exercise:
Show that $K/k$ is purely inseparable $\Leftrightarrow A/J(A) \cong k$, where $J(A)$ is the Jacobson radical of $A$.
It ...
5
votes
1answer
586 views
primitive root of a finite field
This is a problem similar to one of my homework problems, but not on the homework. The problem states that:
Find a primitive root $\beta$ of $F_2[x]/(x^4+x^3+x^2+x+1)$.
Questions:
I know what a ...
5
votes
2answers
678 views
Splitting field of $x^{n}-1$ over $\mathbb{Q}$
From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8
Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...
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votes
2answers
867 views
Degree of $\sqrt{2}+\sqrt[3]{5}$ over $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{5})$
I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, ...
5
votes
2answers
45 views
Galois Extensions and $n^{\text{th}}$ Roots
I've been studying for my prelims lately, and this problem has me stuck:
(a) Let $K$ be a field with no abelian Galois extensions. Suppose that $n$ is a positive integer and either $char(K)=0$ or ...
5
votes
1answer
86 views
A question on morphisms of fields
Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields?
My guess is yes, because every morphism ...
5
votes
1answer
139 views
Calculating Splitting Field Degree of Extension
Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
5
votes
2answers
207 views
Determining subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ without Galois theory?
Somehow I had convinced myself recently that one could determine the subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (here $\zeta_3 = \frac{-1+\sqrt{3}i}{2}$ is a primtive $3$rd root of $1$) without ...
5
votes
2answers
234 views
A slick proof that a field that is finitely generated as a ring is finite
It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,...,x_n]\to k$ for some $n\in \mathbb{N}$, ...
5
votes
1answer
194 views
Integral closure of p-adic integers in maximal unramified extension
Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
