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Author: lenis2000

graduate/exams/algebra/2006-08.html

@@ -225,9 +225,6 @@ <h2 class="unnumbered" id="hints">Hints</h2>
 <li><p>Problems are not sorted according to difficulty.</p></li>
 </ol>
 <h2 class="unnumbered" id="room-for-your-pledge">Room for your pledge:</h2>
-<div class="center">
-<p><img src="images/2025_11_13_81df1e807797aba84082g-02.jpg" alt="Pledge signature box" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
-</div>
 <h2 class="unnumbered" id="problem-1-5-points">Problem 1 [5 points]</h2>
 <p>Prove that there is no simple group of order
 <math role="math" aria-label="2 \times 3^{3} \times 5^{2}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>&times;</mo><msup><mn>3</mn><mn>3</mn></msup><mo>&times;</mo><msup><mn>5</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2 \times 3^{3} \times 5^{2}</annotation></semantics></math>.</p>
@@ -241,13 +238,11 @@ <h2 class="unnumbered" id="problem-2-5-points">Problem 2 [5 points]</h2>
 actually has to be nilpotent; that is, however, a little harder to
 show.) There is partial credit for proving that
 <math role="math" aria-label="G" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>G</mi><annotation encoding="application/x-tex">G</annotation></semantics></math>
-is not simple unless it is Abelian.<br />
-<img src="images/2025_11_13_81df1e807797aba84082g-04.jpg" alt="Work area for Problem 2" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+is not simple unless it is Abelian.</p>
 <h2 class="unnumbered" id="problem-3-5-points">Problem 3 [5 points]</h2>
 <p>Show that any nilpotent group
 <math role="math" aria-label="G" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>G</mi><annotation encoding="application/x-tex">G</annotation></semantics></math>
-of order 900 is Abelian.<br />
-<img src="images/2025_11_13_81df1e807797aba84082g-05.jpg" alt="Work area for Problem 3" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+of order 900 is Abelian.</p>
 <h2 class="unnumbered" id="problem-4-5-points">Problem 4 [5 points]</h2>
 <p>Decide whether</p>
 <p><math role="math" aria-label="x y^{2}+x^{2} y+2 x y+y+x+1" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x y^{2}+x^{2} y+2 x y+y+x+1</annotation></semantics></math></p>
@@ -278,8 +273,7 @@ <h2 class="unnumbered" id="problem-5-5-points">Problem 5 [5 points]</h2>
 <math role="math" aria-label="I" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>I</mi><annotation encoding="application/x-tex">I</annotation></semantics></math>
 but does not contain
 <math role="math" aria-label="a" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">a</annotation></semantics></math>.
-(Hint: use Zorn’s lemma)<br />
-<img src="images/2025_11_13_81df1e807797aba84082g-07.jpg" alt="Work area for Problem 5" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p></li>
+(Hint: use Zorn's lemma)</p></li>
 </ol>
 <h2 class="unnumbered" id="problem-6-5-points">Problem 6 [5 points]</h2>
 <p>Over
@@ -295,13 +289,9 @@ <h2 class="unnumbered" id="problem-6-5-points">Problem 6 [5 points]</h2>
 1 &amp; 2 &amp; -3 \\
 1 &amp; 1 &amp; -2
 \end{array}\right)</annotation></semantics></math></p>
-<div class="center">
-<p><img src="images/2025_11_13_81df1e807797aba84082g-08.jpg" alt="Work area for Problem 6" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
-</div>
 <h2 class="unnumbered" id="problem-7-5-points">Problem 7 [5 points]</h2>
 <p>Determine the number of monic irreducible polynomials of degree 2 in
-<math role="math" aria-label="\mathbb{F}_{7}[x]" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>𝔽</mi><mn>7</mn></msub><mo stretchy="false" form="prefix">[</mo><mi>x</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_{7}[x]</annotation></semantics></math>.<br />
-<img src="images/2025_11_13_81df1e807797aba84082g-09.jpg" alt="Work area for Problem 7" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+<math role="math" aria-label="\mathbb{F}_{7}[x]" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>𝔽</mi><mn>7</mn></msub><mo stretchy="false" form="prefix">[</mo><mi>x</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_{7}[x]</annotation></semantics></math>.</p>
 <h2 class="unnumbered" id="problem-8-5-points">Problem 8 [5 points]</h2>
 <p>Let
 <math role="math" aria-label="M / K" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">M / K</annotation></semantics></math>

graduate/exams/algebra/2007-01.html

@@ -235,8 +235,7 @@ <h2 class="unnumbered" id="problem-2-5-points">Problem 2 [5 points]</h2>
 <math role="math" aria-label="\neq 2" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&ne;</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\neq 2</annotation></semantics></math>.
 Show that any field extension
 <math role="math" aria-label="M / L" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>/</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">M / L</annotation></semantics></math>
-of degree 2 is Galois.<br />
-<img src="images/2025_11_13_c917d5999db6656b490eg-04.jpg" alt="Lattice diagram for Problem 2" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+of degree 2 is Galois.</p>
 <h2 class="unnumbered" id="problem-3-5-points">Problem 3 [5 points]</h2>
 <p>Let
 <math role="math" aria-label="M / L" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>/</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">M / L</annotation></semantics></math>

graduate/exams/algebra/2007-08.html

@@ -230,8 +230,7 @@ <h2 class="unnumbered" id="problem-1-5-points">Problem 1 [5 points]</h2>
 <p>Prove or disprove: the alternating group
 <math role="math" aria-label="\mathbf{A}_{2007}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>𝐀</mi><mn>2007</mn></msub><annotation encoding="application/x-tex">\mathbf{A}_{2007}</annotation></semantics></math>
 has a generating set that consists entirely of elements of order
-2007.<br />
-<img src="images/2025_11_13_7036ed1e19eb96d500a6g-03.jpg" alt="Work space for Problem 1" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+2007.</p>
 <h2 class="unnumbered" id="problem-2-5-points">Problem 2 [5 points]</h2>
 <p>Let
 <math role="math" aria-label="M / K" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>/</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">M / K</annotation></semantics></math>
@@ -318,8 +317,7 @@ <h2 class="unnumbered" id="problem-6-5-points">Problem 6 [5 points]</h2>
 <math role="math" aria-label="n \in \mathbb{N}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&isin;</mo><mi>&Nopf;</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>
 so that
 <math role="math" aria-label="\mathbb{Z}[i n]" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&Zopf;</mi><mo stretchy="false" form="prefix">[</mo><mi>i</mi><mi>n</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[i n]</annotation></semantics></math>
-is not a PID.<br />
-<img src="images/2025_11_13_7036ed1e19eb96d500a6g-08.jpg" alt="Work space for Problem 6" style="max-width: 100%; height: auto; display: block; margin: 1em auto;" /></p>
+is not a PID.</p>
 <h2 class="unnumbered" id="problem-7-5-points">Problem 7 [5 points]</h2>
 <p>Show that</p>
 <p><math role="math" aria-label="p(x)=x^{3}-3 x^{2}+15 x-7" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>15</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">p(x)=x^{3}-3 x^{2}+15 x-7</annotation></semantics></math></p>

graduate/exams/algebra/2015-08.html

@@ -204,20 +204,19 @@ <h1>Algebra general exam</h1>
 receive no help on this exam including from books, notes, internet, etc.
 Put your name on the first page, and initials on all the other pages.
 Good luck.</p>
-<ol>
 <h2 class="unnumbered" id="problem-1">Problem 1</h2>
-<li><p>(10 pts) Find all maximal ideals of
+<p>(10 pts) Find all maximal ideals of
 <math role="math" aria-label="\mathbb{Z}[i]" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&Zopf;</mi><mo stretchy="false" form="prefix">[</mo><mi>i</mi><mo stretchy="false" form="postfix">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[i]</annotation></semantics></math>
-which contain 182 . Find minimal generators for these ideals.</p></li>
+which contain 182 . Find minimal generators for these ideals.</p>
 <h2 class="unnumbered" id="problem-2">Problem 2</h2>
-<li><p>( 10 pts ) Let
+<p>( 10 pts ) Let
 <math role="math" aria-label="r_{1}, r_{2}, r_{3}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>,</mo><msub><mi>r</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">r_{1}, r_{2}, r_{3}</annotation></semantics></math>
 be the roots of the cubic polynomial
 <math role="math" aria-label="X^{3}+10 X^{2}-5 X+4" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>X</mi><mn>3</mn></msup><mo>+</mo><mn>10</mn><msup><mi>X</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>X</mi><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">X^{3}+10 X^{2}-5 X+4</annotation></semantics></math>.
 Find the cubic polynomial with rational coefficients whose roots are
-<math role="math" aria-label="r_{1}^{2}, r_{2}^{2}, r_{3}^{2}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>r</mi><mn>3</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">r_{1}^{2}, r_{2}^{2}, r_{3}^{2}</annotation></semantics></math>.</p></li>
+<math role="math" aria-label="r_{1}^{2}, r_{2}^{2}, r_{3}^{2}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup><mo>,</mo><msubsup><mi>r</mi><mn>3</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">r_{1}^{2}, r_{2}^{2}, r_{3}^{2}</annotation></semantics></math>.</p>
 <h2 class="unnumbered" id="problem-3">Problem 3</h2>
-<li><p>(20 pts)<br />
+<p>(20 pts)<br />
 a) Let
 <math role="math" aria-label="G" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>G</mi><annotation encoding="application/x-tex">G</annotation></semantics></math>
 be a group of order
@@ -242,8 +241,7 @@ <h2 class="unnumbered" id="problem-3">Problem 3</h2>
 such that
 <math role="math" aria-label="x^{2}=1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^{2}=1</annotation></semantics></math>
 for all
-<math role="math" aria-label="x \in G" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>&isin;</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">x \in G</annotation></semantics></math>.</p></li>
-</ol>
+<math role="math" aria-label="x \in G" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>&isin;</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">x \in G</annotation></semantics></math>.</p>
 <h2 class="unnumbered" id="problem-4">Problem 4</h2>
 <p>( 15 pts). Find the characteristic polynomial, the minimal
 polynomial, and the Jordan canonical form of the matrix (over the

graduate/exams/algebra/2021-08.html

@@ -323,7 +323,7 @@ <h2 class="unnumbered" id="problem-4">Problem 4</h2>
 generated by
 <math role="math" aria-label="R" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>
 and
-<math role="math" aria-label="\left.\frac{1}{f}\right)" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>f</mi></mfrac><mo stretchy="true" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\left.\frac{1}{f}\right)</annotation></semantics></math>.
+<math role="math" aria-label="\frac{1}{f})" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>f</mi></mfrac><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{f})</annotation></semantics></math>.
 Prove that</p>
 <p><math role="math" aria-label="R_{f}^{\times} \cong R^{\times} \times \mathbb{Z}^{m} \text { for some } m \in \mathbb{N}" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>R</mi><mi>f</mi><mo>&times;</mo></msubsup><mo>≅</mo><msup><mi>R</mi><mo>&times;</mo></msup><mo>&times;</mo><msup><mi>&Zopf;</mi><mi>m</mi></msup><mrow><mspace width="0.333em"></mspace><mtext mathvariant="normal"> for some </mtext><mspace width="0.333em"></mspace></mrow><mi>m</mi><mo>&isin;</mo><mi>&Nopf;</mi></mrow><annotation encoding="application/x-tex">R_{f}^{\times} \cong R^{\times} \times \mathbb{Z}^{m} \text { for some } m \in \mathbb{N}</annotation></semantics></math></p>
 <p>(Here