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
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1answer
254 views

Methods to show polynomials are irreducible

I would like to show that $x^3 + x^2 - 2x - 1$ is an irreducible polynomial over $\mathbb{Q}$. What are my standard lines of attack to solve this problem? Typically I go to Eistenstein, but it does ...
4
votes
1answer
50 views

Comparing fields with same degree

Two part question: Are the fields $\mathbb{Q} (\sqrt[3]{2}, i \sqrt{3})$ and $\mathbb{Q} (\sqrt[3]{2}, i, \sqrt{3})$ identical in algebraic structure? I have in notes that they both have degree of 6 ...
0
votes
2answers
145 views

Is any homomorphism between two isomorphic fields an isomorphism?

Is any homomorphism between two isomorphic fields an isomorphism? What I mean is that two fields are called isomorphic if there exist one homomorphism between them . But not ...
3
votes
2answers
98 views

If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map

This is from a past qualifying exam. Here is the question: If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map. Here is ...
1
vote
1answer
105 views

Is $C_2$ the correct Galois Group of $f(x)= x^3+x^2+x+1$?

Let $\operatorname{f} \in \mathbb{Q}[x]$ where $\operatorname{f}(x) = x^3+x^2+x+1$. This is, of course, a cyclotomic polynomial. The roots are the fourth roots of unity, except $1$ itself. I get ...
1
vote
1answer
126 views

What is the difference between multiplicative group of integers modulo n and a Galois Field

What is the difference between multiplicative group of integers modulo n and a Galois Field? Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$? Or is it the same as $\mathbb{Z}/n\mathbb{Z}$? ...
5
votes
1answer
75 views

Suppose that characteristic $F$ is $p$. If $K/F$ is separable then $K = F(K^{p})$ where $K^{p} = \{ x^{p} \, |\, x\in K\}$.

I am having difficulty finishing this problem. So far I have this: Want to show $K \subset F(K^{p})$. Since $K/F$ is separable then $K/F$ is algebraic. In particular, $\alpha\in K$ is separable ...
5
votes
1answer
149 views

Multiplicative Group of a Field

The multiplicative group $F^{\times}=F\setminus \{0\}$ of a field is abelian, and it may contain torsion elements, may contain torsion free elements, or both may occur, as can be seen from the ...
3
votes
1answer
388 views

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group.

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group. This is what I did, but I'm not sure if it's right: First, we look at the additive ...
1
vote
2answers
51 views

Extrapolating an abstract algebra proof, arriving upon an incorrect conclusion.

Could you kindly point out what is wrong with my reasoning? EDIT: What I have unintendedly proven through my reasoning is that every field can only have one automorphism- the identity mapping. Hope ...
1
vote
6answers
110 views

Spliting Field over $\mathbb{F}_3$

How to find the splitting field of $f(x)=x^3-x+1$ and $g(x)=x^3-x-1$ over $\mathbb{F}_3$ and how to construct a isomorphism between them?
2
votes
2answers
152 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
4
votes
0answers
70 views

Transcendental elements in $k[[x]]$ over the field $k(x)$

I have a hard time to prove that $k[[x]]$ contains an element which is transcendental over $k(x)$. Could you please explain the some idea how to do that?
1
vote
2answers
174 views

How many field homomorphisms?

Let $F$ and $F′$ be two finite fields with nine and four elements respectively. How many field homomorphisms are there from $F$ to $F′$?
5
votes
3answers
527 views

Prove that a polynomial of degree $d$ has at most $d$ roots (without induction)

Let $p(x)$ be a non-zero polynomial in $F[x]$, $F$ a field, of degree $d$. Then $p(x)$ has at most $d$ distinct roots in $F$. Is it possible to prove this without using induction on degree? If ...
0
votes
2answers
155 views

finitely generated subfield of algebraic closure of the finite field with $p$ elements

Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements. I know that any finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ is ...
3
votes
0answers
56 views

Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
3
votes
2answers
111 views

Can two polynomials $p(x)$ and $q(x)\in \mathbb{F}[x]$ have just one factor in common?

Let $p(x),q(x)\in \mathbb{F}[x]$. EDIT: $\mathbb{F}$ is a field of $0$ characteristic. Let us suppose there is an element $b\notin \mathbb{F}$ such that $p(b)=q(b)=0$. Then, $p(x)$ and $q(x)$ both ...
1
vote
2answers
100 views

Generalized Rationalization in Finite Radical Field Extensions

In the square root case of a radical extension of, say, $\mathbb{Q}$, we have that $\mathbb{Q}(\sqrt{2}) = \{a + b \sqrt{2} | a, b \in \mathbb{Q} \}$. The only semi-hard axiom to prove is that ...
1
vote
1answer
64 views

Help Understanding Fields

I came across this problem in a Linear Algebra text today: Let $u$ and $v$ be distinct vectors in a vector space $V$ over a field $F$. Prove that $\{u,v\}$ is linearly independent if and only if ...
1
vote
1answer
558 views

Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$

Determine the splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$ Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
2
votes
1answer
61 views

A question about splitting fields.

Let $E_{1}$ and $E_{2}$ be two splitting fields of polynomial $p(x)\in \mathbb{F[x]}$ over $\mathbb{F}$. My textbook has a long proof for proving that $E_{1}$ and $E_{2}$ are isomorphic. But isn't ...
1
vote
0answers
57 views

Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$ and their intersection with $k[x_1,\dots,x_n]$

By this thread, if I have a subfield $k\subseteq F\subseteq k(x_1,\dots,x_n)$, $F$ is of the form $F=k(\phi_1,\dots,\phi_m)$ for some rational functions $\phi_1,\dots,\phi_m\in k(x_1,\dots,x_n)$. But ...
1
vote
0answers
71 views

Characterization of normal extensions

I was wondering if there is a characterization of all $\alpha$ algebraic over $\mathbb{Q}$ such that $\mathbb{Q}(\alpha)$ is a normal extension over $\mathbb{Q}$. Also, is there method to prove that ...
1
vote
1answer
181 views

If $F$ is a finite field of size $q=p^n$ and $b\in K$ is algebraic over $F$ then $b^{q^{m}} = b$ for some $m > 0$

I want to apply a similar type of argument to show that $\alpha^{q} = \alpha$ for all $\alpha\in F$. When we know that the characteristic of $F$ is $p$ and $|F| = q = p^{n}$. But I dont know how to ...
2
votes
3answers
155 views

Questions about $\mathrm{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})$

Consider the extension $\mathbb{Q} \subset\mathbb{Q} (\sqrt{2}, \sqrt{3})$. How many elements are there in $\text{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})?$ Describe all elements in ...
5
votes
1answer
54 views

Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$

I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
1
vote
1answer
54 views

Question about minimal polynomials.

Let $p(x)$ be a minimal polynomial for $a$ for field $F$. This implies it is a monic polynomial of the lowest degree possible such that $p(x)=0$. Why does $p(x)$ have to be irreducible? Why can't it ...
2
votes
2answers
47 views

Find $u\in\mathbb{R}$ such that $\mathbb{Q}(u) = \mathbb{Q}(2^{1/2}, 5^{1/3})$.

I am having trouble finding such a $u$. My instincts at first told me to do the obvious thing and let $u = 2^{1/2}5^{1/3}$ but $u^{2} = \left(2^{1/2}5^{1/3}\right)^{2} = 2\cdot5^{2/3}$ but we want ...
1
vote
2answers
33 views

A confusion regarding the nature of elements in a field extension.

I read this statement in a well-known textbook: "Since $[W:F]\leq mn$, every element in $W$ satisfies a polynomial of degree at most $mn$ over $F$." Let $F[a]$ be a field extension of $F$. Can $f\in ...
3
votes
1answer
68 views

Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?

I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write: $ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $ So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means ...
1
vote
1answer
134 views

A question about the degree of an element over a field extension.

Say $K$ is a field extension of field $F$. If element $b$ is algebraic with degree $n$ over $F$, we know that $[F(b):F]=n$. Why is it that $[K(b):K]\leq n$?
1
vote
2answers
52 views

A doubt regarding nature of $F(a)$, where $F(a)$ is the intersection of all fields containing field $F$ and element $a$.

Proof in Herstein: Let us consider elements like $f_{0}+f_{1}a+f_{2}a^{2}+\dots f_{s}a^{s}$. Here $f_{0},f_{1}\dots f_{s}\in F$. Now consider the quotient field $U$ generated by elements like ...
6
votes
4answers
150 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
0
votes
0answers
57 views

A question about field extensions

I read in Herstein that a field extension $K$ of field $F$ is a vector space over $F$, and its degree over $F$ is the number of base elements. Let $K$ (field extension of $F$) be a vector space over ...
2
votes
0answers
57 views

Field extension

I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$. I know I have to prove that $ug(x)-f(x)$ is ...
0
votes
3answers
51 views

automorphisms and field extension $E$ of $\mathbb{Q}$.

I want a hint. That is all I ask for. The question I am asked to prove is as follows: Let $E$ be an extension field of $\mathbb{Q}$. Show that any automorphisn of $E$ acts as the identity on ...
6
votes
1answer
79 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 ...
3
votes
1answer
184 views

Finite Extensions and Roots of Unity

Two questions; the hint I've been provided is that they are, in fact, related. Prove that a finite extension of $\mathbb{Q}$ contains finitely many roots of unity. What is the largest (finite) ...
2
votes
2answers
27 views

Relationships of Eigenvalues in Algebraic Closure

Suppose that $k$ is a field, and $A \in M_n(k)$ is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not ...
6
votes
0answers
191 views

Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)

I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples ...
2
votes
0answers
129 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
11
votes
0answers
224 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
1
vote
0answers
125 views

$X^n-a \in k[X]$ ,char(k)|n. Multiplicity of roots of irreducible polynomial which devides $X^n-a$.

Let $k$ be a field of $char(k)=p>0$, $f(X)=X^n-a \in k[X](a \neq 0)$, $p|n$. If $g(X)$ is an irreducible polynomial in $k[X]$ and $g(X)|f(X)$, do all the roots of $g(X)$ have the same ...
3
votes
0answers
89 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
0
votes
1answer
280 views

Proving an Integral domain is a field. [duplicate]

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field. This is a Ph.D. entrance question, I recently ...
5
votes
2answers
98 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
98 views

finding fixed field of automorphism

Let $F$ be a field and let $g:F(x) \to F(x)$ be the automorphism which maps $x$ to $x+1$. I need to find the fixed field of this automorphism. So far I know $g$ fixes $F$. I want to use Galois ...
2
votes
1answer
358 views

Galois group of $x^8+2$ over $\Bbb{Q}$

This is what I did to find the Galois group for $x^8+2$: Splitting field: $$K = \Bbb{Q}(\zeta_8, \zeta_{16}2^{1/8})$$ Since $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q}) \cong Aut( \langle\zeta_8\rangle) \cong ...
2
votes
1answer
242 views

Calculating The Galois Group of the Splitting Field of $f=x^3-3$

If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions. Now as this is a splitting field and finite ...