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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2answers
69 views

Figuring out whether a ring is a field

Given a ring, how do you test whether it is a field? What properties would you look at?
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2answers
127 views

Linear map on a finite dimensional vector space over an algebraically closed field of characteristic $p>0$

Let $V$ be a finite dimensional vector space over an algebraically closed field $F$ of finite characteristic $p$. Let $\alpha: V\longrightarrow V$ be a linear operator on $V$, and suppose that there ...
16
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3answers
2k views

A finite field cannot be an ordered field.

I am reading baby Rudin and it says all ordered fields with supremum property are isomorphic to $\mathbb R$. Since all ordered finite fields would have supremum property that must mean none exist. ...
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votes
2answers
2k views

automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
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2answers
241 views

Is there an uncountable proper subfield of $\mathbf{R}$?

Whether there is a uncountable proper subfield of real line $\mathbf{R}$?? Thanks a lot!
2
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1answer
64 views

Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
6
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0answers
429 views

Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
5
votes
1answer
164 views

Homework: No field extension is “degree 4 away from an algebraic closure”

Question: Suppose $[L:K]=4$ and char$K \neq 2$ and $L$ is algebraically closed. Show that there is an intermediate field $M$ such that $[L:M]=2$ and that $X^2 + 1$ splits over $M$. Show that this ...
2
votes
1answer
80 views

Splitting field of irreducible polynomails

Can I have two irreducible polynomials of different degree, having isomorphic splitting fields? The base field does not has to be perfect, . I mean if the base field is perfect, the extension is ...
6
votes
2answers
555 views

Showing that a ring homomorphism from a field to a ring is injective

A similar question like this has been asked here, apologies, but need to clarify something at the end Our homework question was to show that any ring homomorphism $f:K\rightarrow R $ (where K is a ...
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2answers
101 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
0
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1answer
58 views

Normal extension and Embeddings

Suppose $K\subseteq Z\subseteq L\subseteq N$ be fields such that $N$ is normal over $K$. For each $K$ embedding $\sigma\in Emb_K(Z,N)$, is it always possible to extend $\sigma$ to an automorphism of ...
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3answers
59 views

classification of $2$-dimensional field extensions

Let $F$ be a field and $K:F$ be a field extension such that $[K:F]=2$. Then (i). If $Char(F)\neq 2$, then there exists $\alpha\in K^*$, $\alpha\notin F^*$, such that $K=F(\alpha)$ and $\alpha^2\in ...
2
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4answers
316 views

example of Frobenius endomorphisms that is not automorphism

Let $F$ be a field of characteristic $p$ and $F$ has infinitely many elements. Prove that the map $\sigma: \alpha\mapsto \alpha^p$ is a field endomorphism but not necessarily automorphism. How to ...
0
votes
1answer
57 views

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit.

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit, where $\delta$ is the Euclidean function. Is it just since $z$ is not a unit then $\delta(z)>\delta(1)>0?$ Please ...
5
votes
1answer
104 views

$f(X^p)$ irreducible or $p$th power if $f$ irreducible

An exercise in Bourbaki: Let $K$ be a field of characteristic $p>0$ and $f$ irreducible monic polynomial of $K[X]$. Show that in $K[X]$ the polynomial $f(X^p)$ is either irreducible or the ...
4
votes
1answer
245 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
4
votes
2answers
72 views

minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
1
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0answers
45 views

integral domain with a field as a subring [duplicate]

I would like to know if my solution to the following exercise is correct. Let $A$ be an integral domain (with a unit) which has a field $\mathbb K$ as a subring and such that $A$ is a ...
3
votes
2answers
51 views

Basis elements for $Q(t)$ as a $Q(t^2)$-vector space.

Let $\mathbb{Q}(t)$ and $\mathbb{Q}(t^2)$ be the fields of rational functions with $t$ and $t^2$ as indeterminates. Both of these fields are infinite-dimensional. How can I determine the dimension of ...
2
votes
1answer
190 views

Splitting field of $X^n-a$

Show that the splitting field of $X^n-a$ over a field $K$ is $K(\alpha, \zeta_n)$, where $\alpha$ is a $n$-th root of $a$ and $\zeta_n$ is a primitive $n$-root of unity.
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2answers
30 views

Algebraic subfields of uncountable fields [closed]

Does every field of uncountable cardinality contain an algebraically closed subfield?
8
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1answer
113 views

Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
2
votes
1answer
242 views

How to determine the Galois group of irreducible polynomials of degree $3,4,5$

Let $f$ be an irreducible (over $\mathbb{Q}$) polynomial in $\mathbb{Z}[x]$, $\deg (f)=3,4,5$. The Galois group of an irreducible polynomial $f\in \mathbb{Z}[x]$ acts transitively on distinct roots in ...
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2answers
61 views

Finding all $c\in \mathbb{Z}_5$ for which $\mathbb{Z}_5[x]/\langle x^3+2x+c\rangle$ is a field.

Finding all $c\in \mathbb{Z}_5$ for which $\mathbb{Z}_5[x]/\langle x^3+2x+c\rangle$ is a field. I have worked out $0$ is not because it factors to $x(x^2+2)$. I believe that $c=1,2,3,4$ are all ...
0
votes
1answer
114 views

When a function field is a regular extension of the field of coefficients?

Let $A$ be an integral affine $k$-algebra with field of fractions $K$. I am wondering when the extension $K/k$ is regular. In particular, is the following statement correct? $K/k$ is regular ...
2
votes
1answer
127 views

Algebraic degree of a product of two algebraic elements

Suppose $(m,n)=1$ and let $a$ and $b$ be algebraic of degrees $m$ and $n$ respectively over $F$. How to prove that $ab$ is algebraic of degree $mn$? It is easy to prove that $ab$ is algebraic of ...
2
votes
2answers
68 views

Showing that $f(a)=f'(a) = 0$ if and only if $f = (x-a)^2q$.

This was problem that I just cannot figure out: Let $F$ be a field, $a\in F$ and $f\in F[x]$. Show that $f(a)=f'(a)=0$ if and only if $f=(x-a)^2q$ for some $q\in F[x]$. And $f'$ refers to the ...
3
votes
1answer
131 views

Splitting field over $\mathbb{F}_3$

The splitting field of $f(x)=x^8-1$ over $\mathbb{F}_3$ is $\mathbb{F}_{3^d}$ where $d=ord_{(\mathbb{Z}/8\mathbb{Z})^*}(3)=2$. But $f(x)=(x^4+1)(x^4-1)$ and $x^4+1$ is irreducible over ...
2
votes
1answer
53 views

If $\alpha$ = $\beta^q - \beta$ where both $\alpha$ , $\beta $ belongs to $F_q^n$ which is extension of $F_q$

Clearly $\beta$ is a root of $f(x) = x^q - x - \alpha$ and the other roots are its conjugates w.r.t $F_q$ so $f(x)$ splits in $F_q^n$ . But the degree is q so there are q distinct roots and my problem ...
0
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1answer
230 views

Composite Field Extension

Let $K_1$, $K_2$ be two finite extensions of $F$ of degree $m$, $n$ respectively. It is well known that if $(m,n)=1$, then $[K_1K_2:F]=[K_1:F][K_2:F]$. Is the converse true? i.e if ...
5
votes
1answer
124 views

Normal curve after base change (p > 0)

Suppose that $C$ is a normal projective curve over some base field $k$, possibly of positive characteristic. I am wondering to what extent one can modify the base field $k$ while not braking the ...
2
votes
2answers
656 views

Specific way of showing $\Bbb Z[\sqrt{-d}]$ is not a Euclidean Domain when $d>2$

Is it true that if a ring is not a UFD then it's not a Euclidean Domain? I have a ring $R=\mathbb{Z}[\sqrt{-d}]=\{ a+b\sqrt{-d} \mid a,b \in \mathbb{Z} \}$ where $d$ is a square free integer. I want ...
2
votes
2answers
1k views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
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0answers
44 views

Why is it true that $F_{q^n} = F_q(\alpha)$ where $\alpha$ is the primitive element of $F_{q^n}$?

Since $\alpha$ is the primitive element of $F_{q^n}$ then $F_{q^n} = \{0, \alpha, \alpha^2,\cdots, \alpha^{q^{n -2}} , 1\}$. Then how $F_q(\alpha)$ is equivalent to $F_{q^n}$? Because what I ...
1
vote
1answer
113 views

Conditions under which a variety to remains smooth after base change (if p > 0)

Let $k$ be an arbitrary field of positive characteristic and let $V$ be a smooth projective (irreducible) variety over $k$. Suppose that $K/k$ is a field extension such that $V_K:=V\times_{\text{Spec ...
0
votes
2answers
47 views

Does there exist a ring containing $k \times k$ that is algebraically closed with respect to $k[x,y]$?

Let $k$ be a field. We know that there exists a field $\bar{k}$ that is an algebraic closure of $k$ with respect to the polynomial ring $k[x]$. But does there exist a ring containing $k^2$ in which ...
1
vote
1answer
76 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
4
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0answers
167 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
3
votes
1answer
131 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
1
vote
1answer
103 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
0
votes
2answers
132 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
0
votes
2answers
167 views

Perfect field of characteristic $p>0$ which is not an algebraic extension of the prime field

True/False If $K$ is a perfect field of characteristic $p>0$, then is $K$ algebraic over $\mathbb{F}_p$? My guess is no and I try to find a counterexample for this. Can anyone give me some ...
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2answers
400 views

Cardinality of algebraic extensions of an infinite field.

An exercise in Lang's algebra book is: let $k$ an infinite field, and $E$ an algebraic extension of $k$. Then $E$ has the same cardinality as $k$. How can one can prove this?
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3answers
90 views

polynomial over a finite field

Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$ Any ideas how to prove it?
1
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1answer
30 views

Simultaneous irreducibility of minimal polynomials

Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is ...
7
votes
1answer
141 views

Short method to prove the irreducibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$. In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
1
vote
1answer
52 views

Newbie material on field theory

I'm studying non-linear systems on my own. I have a basic idea of field diagrams for linear systems in 2d, although I'm not fully grounded in this. Are there any tutorials or material that you would ...
2
votes
0answers
43 views

Intermediate field of $\Bbb Q(\alpha)$ and $\Bbb Q$ [duplicate]

Let $f$ be an irreducible polynomial of degree 4 over $\Bbb Q$ and $Gal(f)=S_4$. Prove that there isn't nontrivial intermediate field between $\Bbb Q(\alpha)$ and $\Bbb Q$ where $\alpha$ is a root of ...
1
vote
1answer
78 views

function of a unit in a Euclidean domain

Let $R$ be a Euclidean domain and let $u$ be a unit in $R$. If we denote $\delta$ the corresponding function, is it true that $\delta(c)=\delta(uc)$ for every non-zero $c \in R$? I know that an ...