Question

Let
E/F be a field extension, and let α be an element of E that is
algebraic over F.

Let p(x) = irr(α, F) and n = deg p(x).

(a) For f(x) ∈ F[x], let r(x) (∈ F[x]) be the remainder of
f(x) when divided by p(x).

Prove that f(x) +p(x)= r(x)+p(x)in F[x]/p(x).

(b) Prove that if |F| < ∞, then | F[x]/p(x)| = |F|n. (For a
set A, we denote by |A| the number of elements in A.)

Answer #1

a.

By the division algorithm, if a(x)a(x) and b(x)b(x) are
*any* polynomials, and a(x)≠0a(x)≠0, then there exist unique
q(x)q(x) and r(x)r(x) such that

b(x)=q(x)a(x)+r(x),r(x)=0 or deg(r)<deg(a).b(x)=q(x)a(x)+r(x),r(x)=0 or deg(r)<deg(a).

Let b(x)=p(x)b(x)=p(x), and a(x)=x−ca(x)=x−c. Then r(x)r(x) must be constant (since it is either zero or of degree strictly smaller than one), so

b(x)=q(x)(x−c)+r.b(x)=q(x)(x−c)+r.

Now evaluate at x=cx=c.

b.

The field F is naturally embedded in F[x]/hp(x)i by

a 7→ a + hp(x)i for a ∈ F. Thus, we may consider

F[x]/hp(x)i as an extension field of F.

and deg p (x)=n thus F (x)/p (x )= Fn

Let F⊆K⊆E be extension fields. If K is an algebraic extension of
F and let α∈E be algebraic over K. Show that α is also algebraic
over F.

Let E/F be an algebraic extension. Let K and L be intermediate
fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E).
(i) Prove that if the extension K/F is separable then the
extension KL/L is separable.
(ii) Prove that if the extension K/F is normal then the
extension KL/L is normal.
Note: To make things easier for you, you can assume that E/F is
finite (hence all extensions are finite),

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

Let E/F be a finite Galois extension such that Gal(E/F) is
abelian. Prove that for
every intermediate field K, the extension K/F is Galois.

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

Let F be an ordered field. Let S be the subset [a,b)
i.e, {x|a<=x<b, x element of F}. Prove that infimum and
supremum exist or do not exist.

Let E be a field of characteristic p, where p is a prime number.
Show that for all x, y that are elements of E, we have (x + y)^p
=x^p + y^p, and hence by induction, (x + y)^p^n = x^p^n + y^p^n
.

Using field and order axioms prove the following theorems:
(i) 0 is neither in P nor in - P
(ii) -(-A) = A (where A is a set, as defined in the axioms.
(iii) Suppose a and b are elements of R. Then a<=b if and
only if a<b or a=b
(iv) Let x and y be elements of R. Then either x <= y or y
<= x (or both).
The order axioms given are :
-A = (x...

Please show all work if needed.
1.Let E be a set with |E| = 3. What is the cardinality of its
power set? That is, find |P(E)|.
QUESTION 2
Find 15 modulo 6
Find the quoitent q and the remainder r when -25 is divided by
9.
Find |_-278.48_|.
Let A and B be sets with A ={1,2,3,7} and B = {a,q,x} with
f: A -> B, with
f(1)=q, f(2) =a , f(3) =q, f(7) =x.
Is f 1-1?
Let...

Let E be an extension of F, and let S be a subset of
E.IfF(S) is the subﬁeld of E
generated by S over F, in other words, the smallest subﬁeld of E
containing F and S,
describe F(S) explicitly, and justify your characterization.

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