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<li><a href="/">Home</a></li>
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<li aria-hidden="true" style="color: #999;">/</li>
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<li><a href="/graduate/">Graduate</a></li>
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<li aria-hidden="true" style="color: #999;">/</li>
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<li><a href="/graduate/generals/">General Exams</a></li>
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<li aria-hidden="true" style="color: #999;">/</li>
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<li><a href="/graduate/exams/algebra/AlgGento02.html">Algebra Exams 1981-2002</a></li>
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<li aria-current="page" style="color: #666;">Algebra General Exam</li>
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<a href="/graduate/exams/algebra/AlgGento02.html" class="back-button">&larr; Back to Table of Contents</a>
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<main id="main-content">
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<h2 class="unnumbered" id="algebra-general-exam">Algebra General
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Exam</h2>
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<ol>
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<li><p>Show (by Sylow theory) that a group of order 72 cannot be simple.
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List all abelian groups of order 72.</p></li>
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<li><p>Let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>G</mi><annotation encoding="application/x-tex">G</annotation></semantics></math>
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be an infinite group containing an element
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x \neq 1</annotation></semantics></math>
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having only finitely many conjugates. Prove that
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>G</mi><annotation encoding="application/x-tex">G</annotation></semantics></math>
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is not simple.</p></li>
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<li><p>Let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>∈</mo><mi>q</mi><mo stretchy="false" form="prefix">[</mo><mi>x</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">f(x)=x^{4}-x^{2}-2 \in q[x]</annotation></semantics></math>.
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Find the splitting field
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math>
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of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math>
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over
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Q</mi><annotation encoding="application/x-tex">Q</annotation></semantics></math>,
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determine its Galois group, and find all subfields of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p></li>
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<li><p>Let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>⊆</mo><mi>K</mi><mo>⊆</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">k \subseteq K \subseteq L</annotation></semantics></math>
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and let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\alpha \in L</annotation></semantics></math>.
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Assume that
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">K / K</annotation></semantics></math>
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is finite algebraic.<br />
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(a) Show that
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">K / K</annotation></semantics></math>
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separable
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo><mi>/</mi><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">=K(\alpha) / K(\alpha)</annotation></semantics></math>
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separable.<br />
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(b) Show that
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">K / K</annotation></semantics></math>
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normal
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo><mi>/</mi><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\Rightarrow K(\alpha) / K(\alpha)</annotation></semantics></math>
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normal.<br />
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(c) Show that if
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">K / K</annotation></semantics></math>
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is Galois, then so is
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo><mi>/</mi><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">K(\alpha) / K(\alpha)</annotation></semantics></math>
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and compare the Galois groups
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo stretchy="false" form="prefix">(</mo><mi>K</mi><mi>/</mi><mi>k</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">G(K / k)</annotation></semantics></math>
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and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo stretchy="false" form="prefix">(</mo><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo><mi>/</mi><mi>K</mi><mo stretchy="false" form="prefix">(</mo><mi>α</mi><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">G(K(\alpha) / K(\alpha))</annotation></semantics></math>.</p></li>
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<li><p>What are necessary and sufficient conditions on the commutative
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integral domain
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>
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in order that the polynomial ring in one indeterminate,
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo stretchy="false" form="prefix">[</mo><mi>x</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">R[x]</annotation></semantics></math>,
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be<br />
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(a) a P.I.D.?<br />
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(b) a UFD?<br />
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(c) a Noetherian ring?</p></li>
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<li><p>Let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>
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be a commutative ring with unit and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>I</mi><annotation encoding="application/x-tex">I</annotation></semantics></math>
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an ideal of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>.
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Consider the
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>-module
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>/</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">R / I</annotation></semantics></math>.<br />
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(a) Show that if
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>I</mi><annotation encoding="application/x-tex">I</annotation></semantics></math>
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is a prime ideal, then
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>/</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">R / I</annotation></semantics></math>
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is indecomposable.<br />
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(b) Show that
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>/</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">R / I</annotation></semantics></math>
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is a simple module if and ouly if
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>I</mi><annotation encoding="application/x-tex">I</annotation></semantics></math>
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is maximal.</p></li>
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<li><p>Let
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>T</mi><annotation encoding="application/x-tex">T</annotation></semantics></math>
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be a linear transformation of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>ℝ</mi><mn>5</mn></msup><annotation encoding="application/x-tex">\mathbb{R}^{5}</annotation></semantics></math>
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having characteristic polynomial
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>T</mi><mn>5</mn></msup><mo>−</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">T^{5}-T</annotation></semantics></math>.
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Find its possible rational canonical forms over
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ℝ</mi><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>
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and its possible Jordan canonical forms over
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ℂ</mi><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>.</p></li>
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</ol>
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</main>
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</body>
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