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
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
1
vote
1answer
20 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 $Aut(K/k)$. ...
2
votes
2answers
35 views

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

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
0
votes
1answer
36 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 ...
6
votes
2answers
380 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
35 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
312 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 ...
1
vote
1answer
57 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 ...
1
vote
1answer
45 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
votes
2answers
44 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 $F^\times$.
0
votes
0answers
23 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 ...
0
votes
1answer
56 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
0answers
22 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?
1
vote
3answers
75 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
2answers
50 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 $f(x)$ ...
2
votes
3answers
146 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?
0
votes
1answer
15 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 ...
2
votes
1answer
148 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 $n$.
3
votes
1answer
35 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
19 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 ...
3
votes
2answers
131 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 ...
2
votes
4answers
144 views

Show that in a field always $0\ne1$

Suppose that $F$ is a field and prove that $0\ne1$ According to the definition of a field I know that the zero element is different from the one element, but is there a scientific proof for that?
1
vote
1answer
39 views

Decomposition of tensor product into direct sum of fields

If I have tensor product of two fields $V_1\otimes V_2$, what is the general approach to decompose this product into a direct sum of fields? In particular, I have $\bullet\;\Bbb Q(\sqrt 2) ...
2
votes
2answers
45 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 ...
1
vote
2answers
30 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 ...
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 ...
3
votes
0answers
28 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 ...
1
vote
2answers
148 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 $F′$?
-2
votes
3answers
37 views

Does represented ring appear to be a field? [closed]

$\mathbb{R}[x]/(x^{2}+1,x^3-2x^2+x-2)$ Hello! My name is Ramzan! I`m solving this issue!
0
votes
1answer
56 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.
0
votes
1answer
45 views

Proposition about intermediate field extensions

This is a problem from Algebra, Hungerford. Exercise V.5.21. (a) Let $L$ and $M$ be intermediate fields of the extension $K \subset F$, of finite dimension over $K$. Assume that $[LM : K:] = [L : ...
1
vote
0answers
28 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 ...
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 ...
2
votes
1answer
34 views

Separability of a polynomial

I have a non zero polynomial $f\in F[X]$ where $F$ is a field. Let $L$ be a field extension of $F$ so that $f$ splits completely in $L[X]$, so $f(X)=c\prod_{i=1}^n (X-a_i)$ with $c,a_i\in L$. If ...
2
votes
1answer
42 views

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$.

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$. Here $K$ is UFD, $Q$ is its field of fractions, $Q(\beta)$ is algebraic extension of $n>1$ degree of ...
5
votes
1answer
457 views

Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
1
vote
2answers
47 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 ...
2
votes
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 ...
0
votes
2answers
30 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
28 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
43 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 $K(i)$. ...
4
votes
2answers
68 views

Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in GF $2^8$

Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in Galois Field $2^8$? Thanks in advance.
0
votes
1answer
24 views

Completion of a metric space vs. field extension

Given a field $F$ and a metric $\mu$, is the completion of $F$ with respect to $\mu$ always a field? Additionally, is there an algebraic field extension of $F$ that is isomorphic to the completion of ...
1
vote
0answers
27 views

Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
2
votes
1answer
65 views

Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$. Any idea? Thanks
3
votes
2answers
102 views

Multiplicative group of a finite field

Field $\mathbb{F}$ is finite if and only if its multiplicative group $\mathbb{F}^{\times}$ is finitely generated. The "$\Rightarrow$" implication is obvious, but how to prove the otherwise?
1
vote
1answer
49 views

Show that a sequence of fields exists

I do not have a clue how to solve the following problem: Let $K\subseteq L$ be Galois extension of degree $p^n$, where $p$ is prime and $n$ is natural. Show that there exists a sequence of subfields ...