Rings and fields: quotient, factorization and extension arguments
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Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 26 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
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For f=X⁴−X−1, compute the discriminant using multiplication by f′ in Q[X]/(f), in the basis 1,X,X²,X³. Prove 283 is prime and verify f=(X−93)²(X−115)(X−265) modulo 283. Identify precisely which step of the good-prime cycle argument fails. Does this factorization imply f splits over Q or that its Galois group is trivial? Reconcile it with the certificates at 2 and 7.
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Determine the Galois group over Q of g=X⁴−X+1 from explicit certificates modulo 2 and 3. You must prove all irreducibility and squarefreeness claims and the subgroup exclusion. Then decide separately whether its roots are expressible by radicals and whether its splitting field can lie in a tower of quadratic extensions of Q. Supply group-theoretic evidence for both decisions; do not infer that S₄ is nonsolvable or that a root degree of four proves constructibility.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In O=Z[√−5], use the proved class group of order two and unique factorization of ideals to characterize irreducible elements by the prime-ideal factorization of their principal ideals. Prove every factorization of a fixed nonzero nonunit into irreducibles has the same number of factors, although O is not a UFD. Apply the criterion to 6=2·3=(1+√−5)(1−√−5).
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In O=Z[s], s²=−5, investigate the rational primes5,7,11. Let A=(s), P=(7,s−3), Q=(7,s+3). Prove (5)=A² and (7)=PQ, calculate all ideal norms, and prove O/(11) is a field. Determine which of A,P,Q are principal and compute P². Explain the different quotient structures O/(5), O/(7), O/(11).
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let s²=−3, R=Z[s], ω=(1+s)/2, and O=Z[ω]. Prove O is the full integer ring. Set J=(2,1+s) as an R-ideal. Show J=2O as subsets of Q(s), calculate [R:J] and [O:J], and compare invertibility in the two rings. Is index multiplicative for J² inside R? Prove whether (2) is prime in O.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In O=Z[s] with s²=−5 let I=(2,1+s), P=(3,1+s), Q=(3,1−s). Reuse the proved I²=(2) and IP=(1+s). Compute the indices of P,Q, prove PQ=(3), find P⁻¹ and P², and determine the relation among their invertible ideal classes. Does this determine the whole class group?
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In Z/18Z, list the units and count the nonzero zero divisors. Describe the ideal generated by 6 and its quotient. Identify all prime ideals and justify whether they are maximal.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Define φ:Z[x]→F₅ by φ(p)=p(2) modulo 5. Prove it is a surjective unital ring homomorphism, determine its kernel, and identify the quotient.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove Z[i] is Euclidean with norm a²+b² using nearest Gaussian integers. Find a gcd of 7+5i and 3+i and a Bézout identity. Explain how the Euclidean proof leads to unique factorization rather than only a numerical gcd.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove x⁴+6x²+3 is irreducible over Q using Gauss and Eisenstein. Explain why absence of rational roots would not suffice, using x⁴+5x²+4. State why the ordinary fraction-field construction applies to Z but not Z/6Z.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct the isomorphism Z/45Z→Z/9Z×Z/5Z and its inverse using idempotents. Solve x≡2 mod 9 and x≡3 mod 5. Contrast with x≡0 mod 4 and x≡1 mod 6.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Three complex numbers, counted with multiplicity, have power sums p₁=0,p₂=2,p₃=6. Recover their monic cubic over Q, prove irreducibility, compute its discriminant and determine its splitting-field Galois group and sign-kernel fixed field. Then explain why reducing the same Newton procedure to characteristic three does not recover the constant coefficient from the first three power sums; exhibit two different cubics with identical such data.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Over F₂ compare f=t²+t+1 and g=t²+1. For each, compute the discriminant and determinant of multiplication by the derivative on its quotient algebra. Determine separability and the splitting-field Galois group. Explain precisely which discriminant implication remains valid and which alternating-group implication fails.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let A=Z[s] with s²=−5 and I=(2,1+s). Using the proved identity I²=(2), identify A/I² as a polynomial quotient over F₂. Determine every ideal of that quotient, its prime and maximal ideals, and the corresponding ideals of A containing (2). Explain how A can be Noetherian while this example has nonunique element factorization and a nonprincipal ideal.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In R=(Z/4Z), let J=(2t,t²) in R[t]. Compute the coefficient ideals Cₙ consisting of coefficients of tⁿ in members of J of degree at most n. Use their stabilization to explain the Hilbert-basis reduction here. Describe all elements and units of B=R[t]/J, and determine its prime and maximal ideals. State why zero divisors do not invalidate the Noetherian conclusion.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Over any field F let A=[[a,1],[0,a]], u=t−a and M=tI−A. Verify U=[[-1,0],[u,1]], V=[[0,1],[1,u]] give a polynomial Smith certificate. Identify the invariant factor and construct a constant cyclic basis P from e₂ and Ae₂. Check the companion matrix and explain why U(t) is not itself the required constant similarity matrix.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let J_k be the nilpotent k-by-k block with ones immediately above the diagonal. Compare A=J₃⊕J₃ and B=J₃⊕J₂⊕J₁ over a field F. Compute characteristic and minimal polynomials, invariant factors and kernel dimensions. Prove that equal characteristic and minimal polynomials need not imply similarity, while explaining why these answers do not conflict with rational canonical classification.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Define φ:Z³→Z/6Z⊕Z by φ(a,b,c)=([2a+3b],c). Prove surjectivity and determine its kernel with an integer basis. Give a finite column-relation presentation of its target, justify the quotient map and compare it with a diagonal four-generator presentation containing a redundant unit relation.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Starting with A=diag(4,6), verify U=[[1,1],[3,2]], V=[[-1,3],[1,−2]] give a Smith certificate. Explain why A itself is diagonal but not Smith form. Classify the quotient, prove the factors using minors, and determine exactly when Ax=b has integer solutions using the transformed system.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A student says that a radical tower is Galois and its group is abelian, so every radical-solvable polynomial has abelian Galois group. Repair the argument completely. Use X³−5 to refute both claims, construct the necessary Galois envelope, and explain the correct obstruction for an independently established S₅ splitting-field group.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let E=F(α,β) be finite separable of degree n over an infinite field F. Show that any set of n(n−1)/2+1 distinct coefficients in F contains a c with E=F(α+cβ). Your proof must justify the bound, including pairs of embeddings agreeing on β. Explain how finite base fields are handled, and give a finite inseparable extension showing why the theorem cannot discard separability.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let α=√2+√7. Determine its minimal polynomial over Q, extension degree and a power basis, proving irreducibility rather than only finding an annihilating polynomial. Express α⁻¹ in that basis.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show √(3+√5) is straightedge-and-compass constructible from a unit segment, using the quadratic-extension mechanism. Prove the real cube root of 5 is not constructible. Explain why a degree that is a power of two is only a necessary test in general.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct F₉ as F₃[t]/(t²+t+2), writing α for the class of t. Prove irreducibility, compute the inverse and multiplicative order of α, and determine Frobenius and its fixed field.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare Q(∛2), the splitting field of x³−2 over Q, and F₃(t)(β) with β³=t. Determine relevant degrees, normality/separability and automorphism counts, proving the distinctions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compute every intermediate field of E=Q(√3,√11) over Q. Give the automorphisms and corresponding subgroups, prove one fixed-field calculation by coefficients, and state why the correspondence is applicable.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For f=X⁴−X−1, compute the discriminant using multiplication by f′ in Q[X]/(f), in the basis 1,X,X²,X³. Prove 283 is prime and verify f=(X−93)²(X−115)(X−265) modulo 283. Identify precisely which step of the good-prime cycle argument fails. Does this factorization imply f splits over Q or that its Galois group is trivial? Reconcile it with the certificates at 2 and 7.
Determine the Galois group over Q of g=X⁴−X+1 from explicit certificates modulo 2 and 3. You must prove all irreducibility and squarefreeness claims and the subgroup exclusion. Then decide separately whether its roots are expressible by radicals and whether its splitting field can lie in a tower of quadratic extensions of Q. Supply group-theoretic evidence for both decisions; do not infer that S₄ is nonsolvable or that a root degree of four proves constructibility.
In O=Z[√−5], use the proved class group of order two and unique factorization of ideals to characterize irreducible elements by the prime-ideal factorization of their principal ideals. Prove every factorization of a fixed nonzero nonunit into irreducibles has the same number of factors, although O is not a UFD. Apply the criterion to 6=2·3=(1+√−5)(1−√−5).
In O=Z[s], s²=−5, investigate the rational primes5,7,11. Let A=(s), P=(7,s−3), Q=(7,s+3). Prove (5)=A² and (7)=PQ, calculate all ideal norms, and prove O/(11) is a field. Determine which of A,P,Q are principal and compute P². Explain the different quotient structures O/(5), O/(7), O/(11).
Let s²=−3, R=Z[s], ω=(1+s)/2, and O=Z[ω]. Prove O is the full integer ring. Set J=(2,1+s) as an R-ideal. Show J=2O as subsets of Q(s), calculate [R:J] and [O:J], and compare invertibility in the two rings. Is index multiplicative for J² inside R? Prove whether (2) is prime in O.
In O=Z[s] with s²=−5 let I=(2,1+s), P=(3,1+s), Q=(3,1−s). Reuse the proved I²=(2) and IP=(1+s). Compute the indices of P,Q, prove PQ=(3), find P⁻¹ and P², and determine the relation among their invertible ideal classes. Does this determine the whole class group?
In Z/18Z, list the units and count the nonzero zero divisors. Describe the ideal generated by 6 and its quotient. Identify all prime ideals and justify whether they are maximal.
Prove Z[i] is Euclidean with norm a²+b² using nearest Gaussian integers. Find a gcd of 7+5i and 3+i and a Bézout identity. Explain how the Euclidean proof leads to unique factorization rather than only a numerical gcd.
Prove x⁴+6x²+3 is irreducible over Q using Gauss and Eisenstein. Explain why absence of rational roots would not suffice, using x⁴+5x²+4. State why the ordinary fraction-field construction applies to Z but not Z/6Z.
Three complex numbers, counted with multiplicity, have power sums p₁=0,p₂=2,p₃=6. Recover their monic cubic over Q, prove irreducibility, compute its discriminant and determine its splitting-field Galois group and sign-kernel fixed field. Then explain why reducing the same Newton procedure to characteristic three does not recover the constant coefficient from the first three power sums; exhibit two different cubics with identical such data.
Over F₂ compare f=t²+t+1 and g=t²+1. For each, compute the discriminant and determinant of multiplication by the derivative on its quotient algebra. Determine separability and the splitting-field Galois group. Explain precisely which discriminant implication remains valid and which alternating-group implication fails.
Let A=Z[s] with s²=−5 and I=(2,1+s). Using the proved identity I²=(2), identify A/I² as a polynomial quotient over F₂. Determine every ideal of that quotient, its prime and maximal ideals, and the corresponding ideals of A containing (2). Explain how A can be Noetherian while this example has nonunique element factorization and a nonprincipal ideal.
In R=(Z/4Z), let J=(2t,t²) in R[t]. Compute the coefficient ideals Cₙ consisting of coefficients of tⁿ in members of J of degree at most n. Use their stabilization to explain the Hilbert-basis reduction here. Describe all elements and units of B=R[t]/J, and determine its prime and maximal ideals. State why zero divisors do not invalidate the Noetherian conclusion.
Over any field F let A=[[a,1],[0,a]], u=t−a and M=tI−A. Verify U=[[-1,0],[u,1]], V=[[0,1],[1,u]] give a polynomial Smith certificate. Identify the invariant factor and construct a constant cyclic basis P from e₂ and Ae₂. Check the companion matrix and explain why U(t) is not itself the required constant similarity matrix.
Let J_k be the nilpotent k-by-k block with ones immediately above the diagonal. Compare A=J₃⊕J₃ and B=J₃⊕J₂⊕J₁ over a field F. Compute characteristic and minimal polynomials, invariant factors and kernel dimensions. Prove that equal characteristic and minimal polynomials need not imply similarity, while explaining why these answers do not conflict with rational canonical classification.
Define φ:Z³→Z/6Z⊕Z by φ(a,b,c)=([2a+3b],c). Prove surjectivity and determine its kernel with an integer basis. Give a finite column-relation presentation of its target, justify the quotient map and compare it with a diagonal four-generator presentation containing a redundant unit relation.
Starting with A=diag(4,6), verify U=[[1,1],[3,2]], V=[[-1,3],[1,−2]] give a Smith certificate. Explain why A itself is diagonal but not Smith form. Classify the quotient, prove the factors using minors, and determine exactly when Ax=b has integer solutions using the transformed system.
A student says that a radical tower is Galois and its group is abelian, so every radical-solvable polynomial has abelian Galois group. Repair the argument completely. Use X³−5 to refute both claims, construct the necessary Galois envelope, and explain the correct obstruction for an independently established S₅ splitting-field group.
Let E=F(α,β) be finite separable of degree n over an infinite field F. Show that any set of n(n−1)/2+1 distinct coefficients in F contains a c with E=F(α+cβ). Your proof must justify the bound, including pairs of embeddings agreeing on β. Explain how finite base fields are handled, and give a finite inseparable extension showing why the theorem cannot discard separability.
Let α=√2+√7. Determine its minimal polynomial over Q, extension degree and a power basis, proving irreducibility rather than only finding an annihilating polynomial. Express α⁻¹ in that basis.
Show √(3+√5) is straightedge-and-compass constructible from a unit segment, using the quadratic-extension mechanism. Prove the real cube root of 5 is not constructible. Explain why a degree that is a power of two is only a necessary test in general.
Construct F₉ as F₃[t]/(t²+t+2), writing α for the class of t. Prove irreducibility, compute the inverse and multiplicative order of α, and determine Frobenius and its fixed field.
Compare Q(∛2), the splitting field of x³−2 over Q, and F₃(t)(β) with β³=t. Determine relevant degrees, normality/separability and automorphism counts, proving the distinctions.
Compute every intermediate field of E=Q(√3,√11) over Q. Give the automorphisms and corresponding subgroups, prove one fixed-field calculation by coefficients, and state why the correspondence is applicable.
