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Algebra-0(R,C)Homework 3.pdf
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Algebra-0(R,C)Homework 3.pdf
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PROBLEM SET 3
Due: March 24, 2023
1.
Let
𝑘
be a field. Let
𝐴
be an
𝑚 × 𝑛
matrix with entries in
𝑘
. Suppose that
𝐴
can be transformed
to a row reduced echelon form
𝑅
1
by finitely many elementary row operations. Suppose that
𝐴
can also be transformed to a row reduced echelon form
𝑅
2
by another collection of elementary row
operations. Show that 𝑅
1
= 𝑅
2
.
2.
Let
𝑅
be a ring. An element
𝑓 ∈ 𝑅
is said to be irreducible, if any factor of
𝑓
is either a unit
𝜆
(i.e.,
there exists 𝜆
′
∈ 𝑅 such that 𝜆 · 𝜆
′
= 1), or a unit times 𝑓 . Find all irreducible elements of C[[𝑥]].
3. Classify ideals of C[[𝑥]]. Does the unique factorization theorem hold for C[[𝑥]]?
4.
A purely imaginary number is a complex number whose real part is zero. Let
𝑓 ∈ R[𝑥]
be a
polynomial whose complex roots are all purely imaginary. Show that its derivative
𝑓
′
has at most
one root that is not purely imaginary.
5.
Find the number of quadratic (i.e., degree two) and cubic (i.e., degree three) monic polynomials in
F
𝑝
[𝑥].
[Information. In general, if
𝑁
𝑛
denotes the number of monic irreducible polynomials of degree
𝑑
in
F
𝑝
[𝑥]
, then a theorem of Gauss says that
𝑁
𝑛
= 𝑛
−1
P
𝑑 |𝑛
𝜇(𝑛/𝑑) 𝑝
𝑑
, where
𝜇
is the Möbius
function. Here is an elementary exposition of this formula.]
6.
Let
𝐶 = {(𝑥, 𝑦) ∈ R
2
: 𝑦
2
= 𝑥
2
+ 𝑥
3
}
. Find polynomials
𝑥(𝑡)
and
𝑦(𝑡)
in
R[𝑡]
such that the map
𝑡 ↦→ (𝑥 (𝑡), 𝑦(𝑡)) defines a surjective map from R onto 𝐶.
[Hint. Draw the graph of
𝐶
using e.g., Wolfram Alpha; then consider the projection from the
“node”.]
7. Does there exist non-constant polynomials 𝑥(𝑡), 𝑦(𝑡) in C[𝑡] such that 𝑦(𝑡)
2
= 𝑥(𝑡)
3
+ 𝑥(𝑡)?
8. Consider the system of linear equations
𝑥
1
+2𝑥
2
+𝑥
3
= 1,
2𝑥
1
+3𝑥
2
+(𝜆 + 2)𝑥
3
= 3,
𝑥
1
+𝜆𝑥
2
−2𝑥
3
= 0.
Find 𝜆 ∈ R such that the equation has infinitely many solutions.
9.
Let
𝐴
and
𝐶
be
𝑚 × 𝑛
real matrices. Suppose for any
𝒃 ∈ R
𝑚
, the systems
𝐴𝒙 = 𝒃
and
𝐶𝒙 = 𝒃
have
the same solution set. Show that 𝐴 = 𝐶.
10.
Find a
100 × 100
real matrix
𝐴
, such that all the entries of
𝐴
are nonzero, and its row reduced
echelon form has precisely 99 pivots.
1
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