Introduction to Abstract Algebra Worksheet

1. (10 points) Let R be a commutative ring (with unity!). An element r ∈ R is nilpotent ifrn = 0 for some n ≥ 0.
(a) Find an example of a ring R and a nonzero element r ∈ R which is nilpotent.
(b) Let R be a ring and let N be the set of nilpotent elements in R. Show that N is an
ideal of R.
(c) Show that in the quotient ring R/N the only nilpotent element is 0.
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2. (10 points) Let F16 = Z2 [x]/(x4 + x3 + x2 + x + 1). Let α = x ∈ F16 . This is a finite field
with 24 = 16 elements. We make the identification

F16 = a0 + a1 α + a2 α2 + a3 α3 |ai ∈ Z2
(a) By Corollary 22.11 (which we proved in class), we know that the multiplicative
×
×
group F×
16 is cyclic. Find an element β ∈ F16 which is a generator for F16 .
(b) Find the roots of f (x) = x2 + x + 1 over F16 .
(c) Find a subfield F4 ⊂ F16 which has order 4. (Hint: as we showed in class, such a
subfield is given by the set of roots of x4 − x in F16 . It might help your calculations
to start by finding the irreducible factorization of x4 − x over Z2 ).
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3. (10 points) Let E/F be a field extension. Let α1 , . . . , αn ∈ E be a collection of elements
which are algebraic over F . Suppose that deg αi = di . Show that
[F (α1 , . . . , αn ) : F ] ≤ d1 · . . . · dn
Hint: show that F (α1 , . . . , αn ) is equal to the F -span of the elements α1m1 · . . . · αnmn
where 0 ≤ mi ≤ di − 1.
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4. (10 points) Let F be a field. Let f (x) ∈ F [x] be an irreducible polynomial of degree d,
and let E be the splitting field of f (x). In this question you will think about the possible
degrees of E over F .
(a) Show that d divides [E : F ], and that d = [E : F ] if and only if E = F (α), where α
is any root of f (x) in E.
(b) Find an example of a field F and an irreducible polynomial f (x) ∈ F [x] such that
d < [E : F ]. (You must prove your example has these properties.) (c) Suppose that d = 2. Show that [E : F ] = 2 (Hint: use the fact that f (x) = (x − α)(x − β) is in F [x]).1 1 Remark: It turns out that in general we have the bound [E : F ] ≤ d!. This is a bit tricky to prove. Page 5 5. (10 points) Let α ∈ C be a number which is algebraic over Q. Suppose that the minimal polynomial f (x) of α over Q has degree 3, and that the splitting field E of f (x) also has degree 3 over Q. Show that α is a real number. Hint: you may use without proof the fact that every degree 3 polynomial over R has a real root (this follows from the intermediate value theorem). Page 6