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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0
votes
1answer
324 views

Smallest subfield containing $F$ and $\alpha$ [duplicate]

Let $F$ be a field and let $K$ be an extension of $F$. Show that if $\alpha\in K$ is algebraic over $F$, $F[\alpha]=\{p(\alpha)\mid p(x)\in F[x]\}$ is the smallest subfield of $K$ containing $F$ and ...
0
votes
3answers
109 views

About isomorphism of rings and fields

If $A,B$ are rings and $A$ is a field. If $A$ is a field and $A\cong B$ so $B$ is a field too? Thank you!
1
vote
2answers
784 views

composition of field extensions

Let fields $K\subseteq L\subseteq M$. Then we know that if $L$ is a finite extension of $K$ and $M$ a finite extension of $L$, then $M$ is a finite extension of $K$. Can we generalize this property? ...
24
votes
6answers
1k views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
8
votes
1answer
155 views

Proving $f = x^8+x^7+x^3+x+1$ to be irreducible over $\mathbf{F}_2$.

I am taking all the irreducible polynomials over $\mathbf{F}_2$ of degree 1 to 4 inclusive and using the long division to determine whether the given $f$ is divisible by anyone of them and it is not ...
1
vote
1answer
51 views

Finding the order of an irreducible polynomial $f$ in $F_3[x]$ of degree 4?

The technique I am using is based on the long division of $x^e - 1$ (e is to be the order) which is really tiresome. So what the other methods (efficient)?
1
vote
2answers
87 views

$[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]\mid [\mathbb{Q}(\beta):\mathbb{Q}]$?

Given $\alpha, \beta$ algebraic numbers over $\mathbb{Q}$, it is known that $d=[\mathbb{Q}(\alpha, \beta):\mathbb{Q}(\alpha)]\le[\mathbb{Q}(\beta):\mathbb{Q}]=b$. It is also true that $d\mid b$ ? If ...
2
votes
4answers
100 views

One question about additive identity arising from Apostol's field axiom in the Mathematical Analysis 2nd edition

In the begining of this book, the field axiom 4 talks about "Given any two real numbers x and y, there exists a real number z such that x+z=y and this z is denoted by y-x." Therefore, for each x, we ...
2
votes
1answer
102 views

Example of a non-Galois extension with $[L : K] = |Aut(L/K)|$

When an extension of fields $L/K$ is finite, we always have $|\operatorname{Aut}(L/K)| \leq [L : K]$, and if $L/K$ is Galois then $|\operatorname{Aut}(L/K)| = [L : K]$. Is the converse true? Is ...
1
vote
2answers
163 views

prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
2
votes
0answers
42 views

Proving integrality of the coefficients “inside the box”

Consider the (usual) $ABKL$ setting: $A$ is an integral domain with field of fractions $K$, $L/K$ is an algebraic field extension, and $B$ is the integral closure of $A$ in $L$ (we are not assuming ...
2
votes
1answer
92 views

Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements

I would like to know if my proof below is correct. Problem Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements. Solution If $\mathbb{F}$ is a ...
6
votes
1answer
173 views

Algebraic and Galois Extension is a Splitting Field of some set.

This is taken from the book Algebra by Thomas W. Hungerford ; Theorem. Let $K$ be an extension of $F$. The following are equivalent: $K$ is algebraic and Galois over $F$. $K$ is separable over $F$ ...
1
vote
2answers
64 views

Composition of Number Fields

If we have number fields $K$ and $L$ containing $\mathbb{Q}$ such that $K\cap L = \mathbb{Q}$, then is it true that $[LK:\mathbb{Q}] = [L:\mathbb{Q}][K:\mathbb{Q}]$?
-1
votes
2answers
293 views

Example of composition of two normal field extensions which is not normal.

If $F\supset E$ and $E\supset k$ are normal extensions. I want a counter-example where $F\supset k$ is not normal
1
vote
1answer
109 views

Splitting a normal extension into a purely inseparable and a separable extension

I would like to prove the following statement. Let $E/F$ be a normal field extension of finite degree and $K=Fix(Gal(E/F))$, then $K/F$ is purely inseparable. (By the way, $E/K$ is Galois, for which ...
2
votes
1answer
518 views

Number of distinct roots of an irreducible polynomial divides the degree of the polynomial

Let $f(X)$ be an irreducible polynomial in the polynomial ring $k[X]$ over the field $k$. Prove that the number of distinct roots of $f(X)$ divides the degree of $f(X)$.
2
votes
1answer
125 views

What power can have irreducible polynomials over a subfield of an algebraically closed field?

$F$ - algebraically closed field. $k$ is a subfield of $F$. The dimension of $F$ over $k$ is a finite number $n$. GENERAL QUESTION: What power can have then irreducible polynomials over $k$? My ...
2
votes
1answer
225 views

Galois group of $X^4+X^3+1$ over $\mathbb{F}_4$

I'm confused. Realizing $\mathbb{F}_4=\mathbb{F}_2[T]/(T^2+T+1)$, the polynomial $X^4+X^3+1$ splits as $(X^2+TX+T)(X^2+(T+1)X+T+1)$. These 2 factors have no root over $\mathbb{F}_4$, so they're ...
4
votes
3answers
397 views

One-to-one homomorphism $f: F\to F$ which is not onto?

The following question comes from a past qualifying exam: What is an example of a ring homomorphism $f: F\to F$ such that $f$ is one-to-one but not onto? (Here $F$ is assumed to be a field.) ...
1
vote
2answers
70 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?
0
votes
2answers
134 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
votes
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. ...
2
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 ...
3
votes
2answers
330 views

Is there an uncountable proper sub-field of $\mathbf{R}$?

Is there an uncountable proper subfield of $\mathbf{R}$?
2
votes
1answer
65 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 ...
7
votes
0answers
474 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
165 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
82 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
674 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 ...
1
vote
2answers
104 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
votes
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 ...
1
vote
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
votes
4answers
366 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
58 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
114 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
261 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
74 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
vote
0answers
46 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
216 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.
0
votes
2answers
33 views

Algebraic subfields of uncountable fields [closed]

Does every field of uncountable cardinality contain an algebraically closed subfield?
8
votes
1answer
115 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
265 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 ...
1
vote
2answers
63 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
138 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
132 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
152 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
54 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 ...