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Introduction to Abstract Algebra I MATH 321 Fall 2025

Worksheet Homework 7

1.

Let \(\varphi:G \to H\) be a group homomorphism with kernel \(K\text{.}\) Prove that \(\varphi\) is injective (i.e., one-to-one) if and only if \(K = \{e\}\) (that is, the only element in the kernel is the identity).
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Hint.
The definition of a function being injective is that every element in the codomain is the image of at most one element of the domain. In other words, if \(g_1 \ne g_2\) are distinct elements of the domain, then \(\varphi(g_1) \ne \varphi(g_2)\) (that is, different inputs must go to different outputs).
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In practice, it is easier to use the contrapositive of this: \(\varphi\) is injective precisely when if \(\varphi(g_1) = \varphi(g_2)\text{,}\) then \(g_1 = g_2\text{.}\) Using this “definition” of injective is what you want to do for this exercise.
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2.

Find an example of rings \(R\) and \(S\) and a function \(\varphi:R \to S\) that is a group homomorphism but not a ring homomorphism.
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Hint.
In the previous homework set, you found a subring of \(\Z_3 \times \Z_3\) that is not an ideal (but of course, it is a normal subgroup).
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3.

Let \(G\) be an abelian group. Let \(H = \{g^2 \st g \in G\}\) and \(K = \{g \in G \st g^2 = e\}\text{.}\)
  1. Find \(H\) and \(K\) for a particular example: the group \(G = U(7) = \{1, 2, 3, 4, 5, 6\}\) where the operation is multiplication mod 7. (You might want to write out the group table first.)
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  2. Use the Fundamental Homomorphism Theorem the relate the groups \(H\) and \(K\text{.}\) That is, find a surjective homomorphism from \(G\) to one of them, such that the other is the kernel. Justify your claims (including that your function really is a homomorphism). Then explain how one is isomorphic to a quotient group involving the other.
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    You can do this for the specific example in part (a), but also make sure you explain why this works in general.
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4.

Consider the ring \(\R[x]\) of polynomials in the variable \(x\) with real coefficients. We can divide polynomials using long division, which results in some remainder.
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(a)

What is the remainder when you divide \(3x^4 - 7x^3 + x^2 - 3x + 7\) by \(x^2 + 1\text{?}\) (Perform the long division, but you only need to submit the results: quotient and remainder.)
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(b)

What do you get when you substitute \(x^2 = -1\) into \(3x^4 - 7x^3 + x^2 - 3x + 7\text{?}\)
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(c)

Coincidence? What sort of polynomial will you always get as a remainder when dividing a polynomial by \(x^2 + 1\text{?}\) What sort of polynomial will you always get when substituting \(x^2 = -1\) into a polynomial?
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(d)

Consider the quotient ring \(\R[x]/\langle x^2 + 1\rangle\text{.}\) Use an evaluation homomorphism (with \(a = \sqrt{-1}\)) to prove that this quotient ring is isomorphic to \(\bC\) (the complex numbers, \(\bC = \{a+bi \st a,b \in \R; i^2 = -1\}\)).
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