Math 454 Spring 2017, PSet 5

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Problem Set 5: Math 454 Spring 2017 Due Thursday February 16

January 6, 2017

Do the problems below. Please write neatly, especially your name! Show all your work and justify all your steps. Write in complete, coherent sentences. I expect and openly encourage you to collaborate on this problem set. I will insist that you list your collaborators on the handed in solutions (list them on the top of your first page). Problem 1. Let E/F be an extension of fields and β ∈ E be algebraic. Prove that if P ∈ F[t] is a non-zero polynomial such that P(β ) = 0, then the minimal polynomial Pβ of β over F divides P. In particular, the minimal polynomial of β is irreducible. Problem 2. Let E/F be a finite extension and L : E → M(n, F) be an injective F–algebra homomorphism with n = [E : F]. (a) Prove that if cL(β ) is the characteristic polynomial of L(β ), then Pβ divides cL(β ) . (b) Prove that there exists an F–basis {β1 , . . . , βn } of E such that for each β ∈ E, if we define the n by n matrix (Aβ ) to have (i, j) coefficient (αi, j ) where αi, j is defined by n

β β j = ∑ αi, j βi , i=1

then L(β ) = Aβ . Problem 3. Let E/F be a finite extension of degree n and m ∈ N with m < n. Prove that there cannot be an injective F–algebra homomorphism L : E → M(m, F). Problem 4. Let P1 , P2 ∈ F[t]. Prove that if P ∈ F[t] divides gcd(P1 , P2 ), then there exists an α ∈ F such that P = α gcd(P1 , P2 ). Problem 5. Let P1 , P2 ∈ F[t]. Prove that hP1 i hP2 i = hgcd(P1 , P2 )i. Problem 6. Let m1 , . . . , mn ∈ N and assume that gcd(mi , m j ) = 1 for all i 6= j. (a) Prove that

n \

mi Z = (m1 . . . mn )Z.

i=1

(b) Prove that

n

Z/ hm1 . . . mn i ∼ = ∏ Z/mi Z. i=1

(c) Prove that if s1 , . . . , sn ∈ N, then

n \ i=1

si Z = lcm(s1 , . . . , sn )Z.

Math 454 Spring 2017, PSet 5

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(d) Prove that if s,t ∈ N and s divides t, then there exists a surjective ring homomorphism ψs,t : Z/tZ → Z/sZ. (e) If s = lcm(s1 , . . . , sn ), prove that there exists an injective homomorphism n

ψ : Z/sZ → ∏ Z/si Z i=1

such that the index of ψ(Z/ lcm(s1 , . . . , sn )Z) in ∏ni=1 Z/si Z is gcd(s1 , . . . , sn ). (f) Deduce that if s1 , . . . , sn ∈ N, then s1 . . . sn = lcm(s1 , . . . , sn ) gcd(s1 , . . . , sn ). Problem 7. Let P1 , . . . , Pn ∈ F[t] and assume that gcd(Pi , Pj ) = 1F for i 6= j. (a) Prove that

n \

hPi i = hP1 . . . Pn i .

i=1

(b) Prove that

n

F[t]/ hP1 . . . Pn i ∼ = ∏ F[t]/ hPi i . i=1

Problem 8. Let F be a field and a be a non-zero, proper ideal in F[t]. Prove that if P1 , P2 ∈ a have minimal degree (among the non-zero elements), then there exists α ∈ F such that αP1 = P2 . In particular, if P1 , P2 ∈ a have minimal degree, then a = hP1 i = hP2 i. Deduce that for each non-zero ideal a, there exists a unique monic polynomial P of minimal degree and hPi = a. Problem 9. Let P1 , . . . , Pn ∈ F[t] and hPi =

n \

hPi i .

i=1

(a) Prove that if Q ∈ F[t] and Pi divides Q for i = 1, . . . , n, then P divides Q. Moreover, there is a unique monic polynomial P with this property. We call such a P the least common multiple of P1 , . . . , Pn and we denote it by lcm(P1 , . . . , Pn ). (b) Prove that there exists α ∈ F such that P1 . . . Pn = α lcm(P1 , . . . , Pn ) gcd(P1 , . . . , Pn ). Problem 10. Let P ∈ F[t]. Define def

dP(t) =

m

∑ iαit i−1

i=1

where

m

P(t) = ∑ αit i . i=0

(a) Prove that D : F[t] → F[t] defined by D(P) = dP is an F–linear function. (b) Prove that P ∈ F[t] is separable if and only if gcd(P, dP) = 1F . (c) Prove that if char(F) = 0, then every irreducible polynomial P ∈ F[t] is separable. [Hint: Use (b)] (d) Prove that if F is a field with char(F) = p 6= 0 and P ∈ F[t] is irreducible and not separable, then there exists Q ∈ F[t] such that P(t) = Q(t p ). [Hint: Use (b)]