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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3
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2answers
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Isomorphic multiplicative groups of quadratic extensions

What are all ordered pairs $(n,m)$ such that the multiplicative groups of the fields $\mathbb{Q}(\sqrt{n})$ and $\mathbb{Q}(\sqrt{m})$ are isomorphic? I saw a question earlier today claiming that ...
4
votes
1answer
56 views

Field extensions of $\prod \Bbb F_p /U$

The ultraproduct of all finite prime fields $ \Bbb F_p $ (over a nonprincipal ultrafilter U) is a field of characteristic 0. How do I show that it has exactly one extension of degree n for each ...
8
votes
4answers
258 views

Why is $\{a + b\sqrt2 + c\sqrt3 : a\in\Bbb{Z}, b, c \in\Bbb{Q}\}$ not closed under multiplication?

The set $R = \{a + b\sqrt{2} + c\sqrt{3}: a \in \Bbb{Z}, c, b \in \Bbb{Q}\}$ is not closed on multiplication, my textbook states. Why is this? And related to that: why then is $S = \{a + b\sqrt{2} : ...
4
votes
1answer
368 views

which of the following is/are algebraic over rationals

which of the following is/are true? $\sin 7^\circ$ is an algebraic over $\mathbb{Q}$ $\sin^{-1}(1)$ is algebraic over $\mathbb{Q}$ $\cos (\pi/7)$ is algebraic over $\mathbb{Q}$ $\sqrt{2}+\sqrt{\pi} ...
2
votes
1answer
59 views

$a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$.

I want to find a $a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$. Any ideas? Thanks.
3
votes
2answers
116 views

Separability of polynomials

From my textbook (Robert Ash's Basic Abstract Algebra, section 3.4): 3.4.2 Proposition If $$f(X)=a_0 + a_1X + \dots + a_nX^n \in F[X],$$ let $f'$ be the derivative of $f$, defined by ...
7
votes
2answers
135 views

Showing $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$

In order to prove $[\mathbb{Q}(2^{1/3}+2^{1/5}):\mathbb{Q}]=15$, I want to show $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$. Any suggestions?
3
votes
1answer
324 views

basic problem about field extension and irreducibility of polynomial

If $\alpha_1, \cdots, \alpha_n$ are distintct roots of a polynomial $p(x)$. If I want to show that $p(x)$ is irreducible over a field $F$, is it suffices to show that $deg (p) \leq [F(\alpha_1, ...
2
votes
2answers
74 views

Proof that field extensions of degree 3 over $K$ with $\mathrm{char}(K) \neq 3$ are separable

My question is how one can prove, for all field extensions $K \subset L$ with $[L:K]=3, $ char($K$) $\not=3$, that $L$ is separable over $K$. I understand this proof with 2 in stead of 3. I ...
8
votes
2answers
1k views

Show that an algebraically closed field must be infinite.

Show that an algebraically closed field must be infinite. Answer If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F ...
0
votes
1answer
532 views

Radical extensions

Suppose we have $$ K \subset K(a_1) \subset K(a_1, a_2) \subset K(a_1, a_2, a_3) $$ such that $a_j^{p_j}\in K_{j-1}$: a radical extension in other words. I am having trouble understanding why for ...
2
votes
1answer
78 views

Prove that this splitting field has degree 4 over $\Bbb{Q}$.

If $m$ and $n$ are distinct square-free positive integers greater than $1$, show that the splitting field $\Bbb{Q}(\sqrt{m}, \sqrt{n})$ of $(X^2-m)(X^2-n)$ has degree 4 over $\Bbb{Q}$. Proof ...
2
votes
1answer
165 views

On the fundamental theorem of field extensions

I'm re-reading the fundamental theorem of field extensions. (K is normal $\iff$ K is a factorization field.) Assume $K=F(\alpha_1, \dots , \alpha_n)$, is the factorization field of $f\in F[x]$, over ...
5
votes
3answers
2k views

Understanding examples of subfield and prime subfield of a finite field

I have already taken a look at this answer. Somehow it did not answer my question. As I can find, in various literatures, A lecture note, Definition 4.1: Let $F$ be a field. A subset $K$ that is ...
1
vote
1answer
109 views

Finite fields as splitting fields

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the ...
0
votes
1answer
64 views

Primitive element (fields)

I'm re-reading the primitive element lemma and I can't reason the following concept. Let $f,g\in F[x]$ be in the polynomial ring of one variable over the field $F$. Let those two polynomials have a ...
3
votes
0answers
95 views

Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see ...
6
votes
3answers
2k views

Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$

Determine all the subfields of the splitting fields of this polynomial. I chose this problem because I think to complete it in great detail will be a great study tool for all of the last chapter, as ...
10
votes
2answers
255 views

This tower of fields is being ridiculous

Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2). Now obviously ...
4
votes
1answer
226 views

Degree of splitting field extensions

The problem states: Let $f (x) = x^3+px+q$ be an irreducible cubic polynomial with rational coefficients and let $K$ be the splitting field of $ f(x) $ over $\mathbb{Q}$. Prove that $ [K : ...
8
votes
3answers
527 views

Do maximal proper subfields of the real numbers exist?

To clarify the problem, consider the field ${\Bbb R}$ as a field extension of ${\Bbb Q}$ using some sort of Hamel basis. Does there exist a field $F\subsetneq{\Bbb R}$ such that $F(\sqrt{2})={\Bbb R}$ ...
7
votes
0answers
98 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
2
votes
1answer
93 views

Non-archimedean matrix fields

Are there any examples of sets $A\subseteq{\Bbb C}^{n\times n}$ for which you can find operations so that $(A,+,\,\cdot\ ,<)$ is a non-archimedean ordered field? I feel like the answer is probably ...
2
votes
1answer
505 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
66 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 ...
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vote
2answers
260 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
111 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
112 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
222 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
93 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
168 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
740 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
52 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
124 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
196 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 ...
5
votes
0answers
79 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
253 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
795 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
276 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
66 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
120 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
125 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
76 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 ...
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votes
1answer
1k 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
69 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 ...
2
votes
0answers
63 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
1answer
249 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
218 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
69 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
64 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 ...