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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5
votes
1answer
57 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
3
votes
0answers
39 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
3
votes
2answers
60 views

Is there a universal property in this proposition (which regards field extensions)?

Since learning a bit of category theory, I am trying, as an exercise, to state results that I come across in categorical language. I am trying to do this with the following: Let ...
3
votes
1answer
47 views

what is this property called?(Field theory)

In what references the following property $P$ of a field $F$ is investigated? The property $P$: For all $n\in \mathbb{N}$ if $\sum_{i=1}^{n} f_{i}^{2}=0,\;\;f_{i}\in F$ then $f_{i}=0, \forall i\in ...
6
votes
2answers
90 views

Euclid and finite fields

In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set. Much ...
0
votes
1answer
48 views

Possible Fields?

Is there an algebraically closed field which is a 1-dimensional vector space (as opposed to complex numbers which are 2-D)? Also is there a complete $\aleph_0$ field?
2
votes
1answer
29 views

If $L_1/K$ and $L_2/K$ are not Galois (solvable), then $L_1L_2/K$ is not Galois (solvable)

This is part of an exam preparation: Prove/contradict: If $L_1/K$ and $L_2/K$ are not Galois, then $L_1L_2/K$ is not Galois. If $L_1/K$ and $L_2/K$ are not solvable Galois extensions, ...
0
votes
2answers
32 views

Is $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$

Is $$\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$$ I know that $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}(i)$ but I can't say that $\mathbb{Q}[X]/(X^2-1) \cong \mathbb{Q}(1) = \mathbb{Q}$ ...
2
votes
0answers
37 views

Why is the subfield (of a field) generated by an algebraic element equal to the subring generated by the same element?

I am trying to prove that, for a field extension $\mathbf{K}/k$ and $a$ an algebraic element over $k$, $$k(a)=k[a],$$ where $k(a)$ is the subfield of $\mathbf{K}$ generated by $a$ and $k[a]$ is the ...
1
vote
1answer
40 views

A field extension of degree 8

I would really appreciate it if you give me a hint on the following question: If $K \subset F$ is a field extension of degree 8, then we must have $F=K(a,b,c)$ for some a, b and c in F.
6
votes
2answers
89 views

Dimension of $\Bbb Q(e)$ over $\Bbb Q$?

The dimension of $\Bbb Q(\sqrt{2})$ over $\Bbb Q$ is finite since $\sqrt2$ is algebraic over $\Bbb Q$. But what about any transcendental number (say $e$)? Which is the smallest field containing $\Bbb ...
2
votes
1answer
24 views

Field Isomorphisms between a field and something that contains it

Are there any k-isomorphism of fields between M and L such that K $\subseteq$ M $\subset$ L? Examples would be appreciated. Thanks
23
votes
6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
2
votes
2answers
66 views

If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field? Find necessary and sufficient condition. Attempt: Since we ...
5
votes
2answers
403 views

Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

$F$ is a field of order $32$. $F$ and {$0,1$} are trivial subfields of $F$. But how can we show that these are the only subfields of $F$? Can someone give me a direction to this question?
0
votes
1answer
38 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
1
vote
1answer
28 views

Normalizer of a subgroup of a Galois group

I wanted to check whether my solution for this problem was correct. Let $k \subseteq L \subseteq K$ be a finite extension of fields, with $K/k$ Galois $H$ the normalizer of $Aut(K/L)$ in $Aut(K/k)$. ...
0
votes
1answer
36 views

A request for a particular example in field theory.

I'm looking for an example of the following kind: Let $a,b\notin \Bbb{Q}$, where $a$ and $b$ satisfy the irreducible polynomials $p(x)$ and $q(x)\in\Bbb{Q}[x]$ respectively. The irreducible ...
9
votes
2answers
319 views

Is there a 'conjugation' on every algebraically closed field?

Let $K$ be an algebraically closed field. Then the polynomial $x^2+1\in K[x]$ has two distinct roots (when $K$ doesn't have characteristic 2). Let's suggestively call them $i$ and $-i$. Does there ...
0
votes
0answers
24 views

Hensels Lemma in many variables

Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$. Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable ...
-1
votes
2answers
45 views

Subgroup of roots of unity of a field. [closed]

Let $F$ be a field. Show that the set of all $n$th roots of $1$ is a subgroup of $F^\times$.
1
vote
1answer
98 views

What is the left adjoint of the forgetful functor from fields to integral domains?

I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain: "There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category ...
0
votes
1answer
50 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
1
vote
1answer
61 views

If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
0
votes
1answer
58 views

Prove $Z_{p}$ are prime fields,where $p$ is prime numbers

show that $Z_{p}$ are prime fields,where $p$ is prime numbers. maybe this problem is old,But I look for some book,and can't find it,someone know which book have this problem proof? because I know ...
1
vote
1answer
59 views

Proof that $a\equiv b \pmod n \iff a \pmod n = b\pmod n$

Proof that for every $a,b \in \mathbb Z,\ n \in \mathbb N$, that $$a\equiv b \pmod n \iff a \pmod n = b \pmod n.$$ My approach is: $n\mid a$ and $n\mid b$ $a\equiv b \pmod n \iff \exists x,y: ...
3
votes
2answers
33 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
2
votes
0answers
23 views

Proving that a field of characteristic $0$ is the field of fractions of a proper subring.

If $K$ is a field of characteristic $0$, $A$ is a subring of $K$ maximal subring of $K$ which doesn't contain $\frac{1}{2}$, and $F$ is the field of fractions of $K$, then I have proved that $K$ is ...
0
votes
1answer
33 views

How solve $[20]_3^{-1}$?

What does this mean, $[20]_3^{-1}$? it's from the topic rings, fields and residue classes. Can you give me a hint how to solve this?
0
votes
2answers
58 views

Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
1
vote
3answers
77 views

$K^\times$ isomorphic to $\mathbb{Z}/2n\mathbb{Z}$ when $K^\times$ is cyclic

Let $K$ be a field so that $K^\times$ is cyclic. Assume $\operatorname{char} K \neq 2$. Prove that $K$ is finite and $K^\times$ is isomorphic to $\mathbb{Z}/2n\mathbb{Z}$ for some $n$. To prove that ...
0
votes
1answer
18 views

Splitting Field of Cubic Polynomial Over the Rationals

I'm having a hard time wrapping my head around some of concepts Pinter's Abstract Algebra introduces about splitting fields (or root fields, as it calls them). Hopefully if I can be pointed in the ...
3
votes
1answer
40 views

Embedding Fields in Matrix Rings

Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb ...
0
votes
2answers
20 views

A question about the minimal polynomials of linear transformations in fields.

Let $K$ be a finite field extension of $F$. Let $\alpha\in K\setminus F$. Then multiplication by $\alpha$ is an F-linear transformation form $K\to K$. Let linear transformation be called ...
2
votes
1answer
155 views

Is $\mathbb{R}^n$ a field?

Is $\mathbb{R}^n$ a field for all $n$? I suppose for n=1 and 2 the result is clear. What about higher values of $n$.
2
votes
3answers
156 views

Why does Fld not have an initial object?

My Algebra book says that the category Fld of fields has no initial object. Why would $\{0,1\}$ not be an initial object? Does it not have a unique homomorphism to every other field?
4
votes
2answers
154 views

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
1
vote
2answers
31 views

Show the following subspaces are invariant

Let $V$ be a vector space over a field $F$ and let $\alpha \in End(V)$. IF $W$ and $Y$ are subspaces of $V$ which are invariant under $\alpha$, show that both $W+Y$ and $W\cap Y$ are invariant under ...
2
votes
2answers
51 views

$x\in K(t)$ is algebraic over $K$ if and only if $x\in K$ (with $t$ transcendental over $K$)

I already proved the following: Lemma A: Let $K$ be a field. Let $t$ be transcendental over $K$. Let $f \in K[X] \backslash K$. Then $f(t)$ is transcendental over $K$. Proof: Because $t$ is ...
3
votes
0answers
67 views

Why do fields seem to be a prerequisite for calculus?

I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation ...
2
votes
1answer
47 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
0
votes
1answer
66 views

How are the fields $\mathbb{F}_k$(where $k$ is an integer) be generated?

What are elements like in the fields $\mathbb{F}_k$? Does $\mathbb{F}_k$ contain only $k$ elements? When $k$ is a composite integer, what will be different from that $k$ is a prime? Please help me.
1
vote
0answers
30 views

intersection of radical extensions of Q

Are there radical extensions $\mathbb{Q}\subseteq R_1$ and $\mathbb{Q}\subseteq R_2$ such that $R_1 \cap R_2$ is not radical over $\mathbb{Q}$? My guess is that one can find a such example, but I ...
3
votes
1answer
51 views

Is $\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$ a simple extension?

Is the extension $$\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$$ simple? If so find the minimal polynomial and the basis for the extension.
2
votes
0answers
35 views

Characterizing quadratic number fields that are subfields of cyclic quartic number fields [duplicate]

Given a quadratic number field $F = \mathbb{Q}(\sqrt{d})$, is there a way to determine whether or not $F \subset K$ for some quartic numberfield $K$ with $\operatorname{Gal}(K/\mathbb{Q}) \cong ...
1
vote
0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
1
vote
2answers
53 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
0
votes
2answers
31 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
0
votes
1answer
31 views

$[K:F_1]=[K:F_2]$, are $F_1, F_2$ isomorphic?

$K$ is a field extension of field $F_1$, $F_2$, all of them are finite fields. There is no other conditions. I want to know is the assertion possible? Could you please give a brief explanation?
1
vote
0answers
44 views

Prove that every sum of squares in $K$ is a square in $K$, where $K$ is certain field.

Let $K$ be a field such that $f(t)=t^{2}+1$ is an irreducible polynomial in $K[t]$. Let $i$ be a root of $f$ in an algebraic closure of $K$. Suppose every element of $K(i)$ is a square in $K(i)$. ...