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 ...

learn more… | top users | synonyms

0
votes
3answers
46 views

clarification of algebraic closure and algebraically closed field

Definition of Algebraic closure: An extension K of F is called an algebraic closure of F if a) F $\subset$ K is algebraic b) K is algebraically closed given the above definition I have been trying ...
3
votes
2answers
42 views

$F(x,y)$ over $F$ is not simple

Let $F$ be a field. Let $x,y$ two algebraically independent indeterminates. Show that $F(x,y)/F$ is not a simple extension. Attempt: I tried by contradiction, assuming that $F(t)=F(x,y)$ and writing ...
2
votes
0answers
53 views

When $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta)$?

Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$ \mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{$\ast$}$$ ...
4
votes
3answers
63 views

$[F(t):F(t^n)]=n$ where $t$ is trascendental

Let $F$ be a field and let $t$ be trascendental over $F$. Prove that $[F(t):F(t^n)]=n$. Obviously $[F(t):F(t^n)]\le n$ since the polynomial $f(x)=x^n-t^n \in F(t^n)$ has $t$ as a root. But I don't ...
2
votes
1answer
66 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients ...
0
votes
0answers
27 views

On the Characterization of $F[\alpha]$ for a Field Extension $E/F$.

Let $E/F$ be a field extension. Reading a proof that $$ \alpha \text{ is algebraic over } F \implies [F[\alpha]:F] < \infty, $$ one might begin by considering an element $f(\alpha) \in F[\alpha]$ ...
2
votes
2answers
75 views

Can the product of only some of the algebraic conjugates be an integer?

Suppose I know that $x_1,\dots, x_n$ are algebraic conjugates and suppose that their product is a rational integer: $$ \prod_{i=1}^{n}x_i\in \mathbb{Z} $$ Is it possible that there exists some other ...
0
votes
2answers
146 views

Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field the element $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers ...
2
votes
1answer
46 views

Proof of a Field Extensions Theorem

Consider the following result. Theorem : Let $E/F$ be a finite field extension of degree $n$ and let $V$ be a vector space over $E$. Then $$ \dim_F V = [E:F] \dim_E V. $$ Now, it seems like a ...
1
vote
1answer
44 views

Algebraic extension of rational numbers.

Let $1<m_1,\ldots,m_r\in{\mathbb{Z}}$. If $K=\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_r})$, and $1<n\in{\mathbb{Z}}$ so that $m_i\nmid{n}$. Is true that $\sqrt{n}\notin{K}$? Added: In addition ...
2
votes
2answers
29 views

A field $K$ is an algebra

I learned this definition of an algebra recently. The definition is: A vectorspace $V$ over a field $K$ is is an algebra if there exists $K$-bilineair map $\varphi\colon V\times V\rightarrow V$ which ...
1
vote
0answers
21 views

If $k$ is a field then $\text{End}_k(k^2)$ is simple

Let $k$ be a field. I have to show that $\text{End}_k(k^2)$ is simple. First of all, I don't see why this is true. For example, if $k=\mathbb{C}(x_1,x_2,\dots)$ then $\varphi\colon k^2\rightarrow ...
0
votes
2answers
35 views

Existence of finite field extension containing a root

I've been thinking about my previous question a bit more, and I'm afraid I still don't quite understand. See: Can the natural embedding $K\to K[X]/(f)$ be extended to form an isomorphism $L/K\to ...
1
vote
1answer
45 views

Calculating Splitting field

Find the splitting field of the polynomial and degree over $\mathbb{Q}$ $P(X)=X^4+2$. The roots of $P(X)$ are $\sqrt[4]{2}\sqrt{i},\ -\sqrt{i}\sqrt[4]{2}, \ i\sqrt{i}\sqrt[4]{2},\ ...
0
votes
0answers
18 views

Minimal polynomial (Field theory)

Let $\alpha$ be the real positive fourth root of 2. My question is: $Polmin(\alpha,\mathbb{Q}(i))=X^4-2$? Because $Polmin(\alpha,\mathbb{Q}(i)) | Polmin(\alpha,\mathbb{Q})=X^4-2$ But ...
2
votes
1answer
32 views

Galois group of order 2^4

Find the galois group the polynomial $f(X)=(X^2-2)(X^2-3)(X^2-5)(X^2-7)$ over $\mathbb{Q}$. A splitting field for $f(X)$ is $K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. We must have ...
2
votes
0answers
24 views

Finite dimensional field extension, finitely many intermediate fields

Good morning, My question is the following: Does every finite dimensional field extension have finitely many intermediate fields? I thought about it quite a while and know that the following is true: ...
1
vote
1answer
38 views

If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

I'm filling the gaps in a proof and I'm stuck in this part: Suppose $R$ is a UFD and $Q$ is a prime ideal of $R[x]$, if $F$ is the quotient field of $R$ and $R\cap Q=\{0\}$, then $QF[x]\cap ...
1
vote
1answer
66 views

Is there a short proof of the formula for Legendre symbol $(\frac{2}{p})=(-1)^{(p^2-1)/8}$?

Let $p$ > 2 be a prime number. I found in wiki a complex proof for this Legendre symbol: $$\left(\frac{2}{p}\right) = (-1)^{\frac{(p^{2}-1)}{8}}$$ Can anyone give me a short solution please?
0
votes
2answers
43 views

Separable extension [closed]

Let $\alpha$ algebraic over $k$ of characteristic $p>0$ Prove that $\alpha$ is separable over $k$ if and only if $k(\alpha)=k(\alpha^p)$. Any suggestion, please.
6
votes
4answers
88 views

Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
1
vote
1answer
31 views

Can the natural embedding $K\to K[X]/(f)$ be extended to form an isomorphism $L/K\to K[X]/(f)$?

I'm studying for an abstract algebra exam (covering commutative rings and Galois theory). As an exercise, I'm trying to work out on my own a proof of the theorem that, given a field $K$ and a ...
-1
votes
1answer
76 views

There are no field structures on $\mathbb{R}^3$, but what of $\mathbb{R}^n$ for $n\geq 4$?

Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$? The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers ...
3
votes
1answer
61 views

Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...
0
votes
3answers
36 views

Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
0
votes
1answer
23 views

Separable polynomial and algebraic extension

If $f\in F[t]$ is separable and $E/F$ is an algebraic extension, then how can I be sure that $f$ is separable as an element of $E[t]$? I thought it is a trivial question...but now I think it is ...
8
votes
1answer
331 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
1
vote
0answers
26 views

Regarding nomenclature of a vector related to field automorphisms

Is there a particular designation in use for the following type of vector, constructed by taking a collection of basis elements $\beta_1$, $\beta_2$, $\dotsc$, $\beta_n$ for a field extension, ...
0
votes
2answers
87 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
3
votes
2answers
64 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x = x(x^{n-1}-1)$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. ...
4
votes
1answer
40 views

Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
1
vote
1answer
37 views

Field extensions: compute the degree of an extension.

I'm stuck with this problem. Let $F\subseteq E$ and $\gamma\in E$ is trascendental over $F$. Let $m$ be a positive integer. Show that $[F(\gamma):F(\gamma^{m})]=m$, where $[\quad:\quad]$ is the ...
0
votes
1answer
46 views

An infinite extension of $\mathbb Q$ [duplicate]

Let $S=\{\sqrt p \in \mathbb R | p $ is a primer number$\}$. How can I show that $\mathbb Q(S)|\mathbb Q$ is an infinite field extension?
4
votes
1answer
37 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
1
vote
0answers
38 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
1
vote
1answer
39 views

Minimal polynomial and field extension

If the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$. Does the above imply ...
0
votes
0answers
20 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
1
vote
1answer
31 views

If the field has prime field isomorphic to $\mathbb{Q}$ would there be any subfield isomorphic to $\mathbb{Q}$?

As title says, if the field has prime field isomorphic to $\mathbb{Q}$ would there be any other subfields isomorphic to $\mathbb{Q}$?
2
votes
1answer
38 views

Splitting field of $(x^2-2)(x^6-20)$ over $\mathbb{Q}$

I have to determine the splitting field $K$ of $f(x)=(x^2-2)(x^6-20)$ over $\mathbb{Q}$. My attempt of solution: $K=\mathbb{Q}(\sqrt2, \sqrt[6]{20}, i\sqrt3)$; $d_1:=[\mathbb{Q}(\sqrt2, ...
1
vote
1answer
34 views

Field of prime characteristic over two indeterminates

Let $F$ have prime characteristic $p$ and let $E = F(Y,Z)$, where $Y, Z$ are indeterminates. Let $L=F(Y^{p} , Z^{p})$ $\subseteq E$. a. Show that $\alpha^{p} \in L$ for all $\alpha \in E$. b. Show ...
0
votes
2answers
53 views

What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
1
vote
1answer
41 views

Proving that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$

if $f(x)$ is a cubic irreducible polynomial over $\mathbb Z_3$, prove that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$ Attempt: $f(x) = \alpha ...
0
votes
0answers
30 views

Presence of non square elements in $GF(p)$

I came across a problem which had a line of explanation as follows : Let $a \in GF(p). a$ is a non square in $GF(p),~ p \neq 2 \implies \nexists~ b\in GF(p)~~|~~a =b^2 $. But, is it really possible ...
1
vote
1answer
53 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
0
votes
0answers
24 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
1
vote
1answer
28 views

Some questions about sub-fields of the field of complex numbers

Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that (1) $F(f)$ contains $f$ as a sub-field and (2) ...
1
vote
0answers
24 views

determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
1
vote
0answers
37 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
3
votes
3answers
116 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
1
vote
1answer
35 views

A field of characteristic zero is perfect

How do you prove that a field F of characteristic zero is perfect, or rather that every irreducible f(x) in F[x] is separable? Thank you!