Dartmouth College
Mathematics 81/111 Homework 3 - solutions
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in
Dartmouth College
Mathematics 81/111 Homework 7
1. Consider the splitting field L of x4 2 over Q. The goal will be to characterize the
Galois group and to write down the complete lattice of subgroups of the Galois group
and the corresponding lattice of in
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Dartmouth College
Mathematics 81/111 Homework 5
1. Let L/K be an extension of fields. We say that K is algebraically closed in L if
the only elements of L which are algebraic over K are the elements of K. Let x be
transcendental over K
Dartmouth College
Mathematics 81/111 Homework 3
1. A first approximation to a theorem of Bezout. It shows that if two curves in the plane are
described by the zero sets of relatively prime polynomials, then the two curves intersect
in only a finite number
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Dartmouth College
Mathematics 81/111 Homework 6
1. Let K be the splitting field over Q of f (x) = (x2 2)(x2 3)(x2 5).
(a) Characterize the Galois group Gal(K/Q), i.e., describe the elements and the isomorphism class of the group.
(b) D
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Dartmouth College
Mathematics 81/111 Homework 4
1. We have seen a number of tests to help determine whether a polynomial is irreducible,
and they will serve you well. On the other hand, neither one size nor one tool fits all
problems.
Dartmouth College
Mathematics 81/111 Homework 2
1. Let Q be the rational numbers.
(a) Let A be the ring Q Q. Determine all the ideals of A, and which among those
are maximal.
(b) In case we have not yet proved it in class, assume that the polynomial ring
Dartmouth College
Mathematics 81/111 Homework 3
1. A rst approximation to a theorem of Bezout.
(a) Let A be a UFD with eld of fractions F . Let f, g A[x]. Show that f, g are
relatively prime in A[x] if and only if f, g are relatively prime in F [x] and (t
Dartmouth College
Mathematics 81/111 Homework 4
1. Let F be a eld of characteristic 0, and let m and n be distinct integers with
n F , and mn F .
/
/
(a) Show that [F ( m, n) : F ] = 4.
m F,
/
(b) Show by example (with m/n (Q 2 ) that the above statement
Dartmouth College
Mathematics 81/111 Homework 1 Solutions
Some basic denitions concerning algebraic sets
Let k be a eld, and let k [x1 , . . . , xn ] the polynomial ring in n variables with coecients
in k . For f k [x1 , . . . , xn ] and P = (a1 , . . . ,
Dartmouth College
Mathematics 81/111 Homework 2
1. Prime and maximal ideals in non-commutative rings. We start with two basic denitions:
Denition: A ring A is simple if its only (2-sided) ideals are cfw_0 and A. We
know that if k is a eld, that k is a si
Dartmouth College
Mathematics 81/111 Homework 1
Some basic definitions concerning algebraic sets
Let k be a field, and let k[x1 , . . . , xn ] the polynomial ring in n variables with coefficients
in k. For f k[x1 , . . . , xn ] and P = (a1 , . . . , an )